$$20(s+2)(s+3)(s+6)(s+8) = 0$$

$$ \begin{matrix}s_1 = -2 & s_2 = -3 & s_3 = -6 \\[1 em] s_4 = -8 & \\[1 em] \end{matrix} $$

$ \color{blue}{ 20x^{4}+380x^{3}+2480x^{2}+6480x+5760 } $ is a polynomial of degree 4. To find zeros for polynomials of degree 3 or higher we use Rational Root Test.

The Rational Root Theorem tells you that if the polynomial has a rational zero then it must be a fraction $ \dfrac{p}{q} $, where p is a factor of the trailing constant and q is a factor of the leading coefficient.

The factors of the leading coefficient ( 20 ) are 1 2 4 5 10 20 .The factors of the constant term (5760) are 1 2 3 4 5 6 8 9 10 12 15 16 18 20 24 30 32 36 40 45 48 60 64 72 80 90 96 120 128 144 160 180 192 240 288 320 360 384 480 576 640 720 960 1152 1440 1920 2880 5760 . Then the Rational Roots Tests yields the following possible solutions:

$$ \pm \frac{ 1 }{ 1 } , ~ \pm \frac{ 1 }{ 2 } , ~ \pm \frac{ 1 }{ 4 } , ~ \pm \frac{ 1 }{ 5 } , ~ \pm \frac{ 1 }{ 10 } , ~ \pm \frac{ 1 }{ 20 } , ~ \pm \frac{ 2 }{ 1 } , ~ \pm \frac{ 2 }{ 2 } , ~ \pm \frac{ 2 }{ 4 } , ~ \pm \frac{ 2 }{ 5 } , ~ \pm \frac{ 2 }{ 10 } , ~ \pm \frac{ 2 }{ 20 } , ~ \pm \frac{ 3 }{ 1 } , ~ \pm \frac{ 3 }{ 2 } , ~ \pm \frac{ 3 }{ 4 } , ~ \pm \frac{ 3 }{ 5 } , ~ \pm \frac{ 3 }{ 10 } , ~ \pm \frac{ 3 }{ 20 } , ~ \pm \frac{ 4 }{ 1 } , ~ \pm \frac{ 4 }{ 2 } , ~ \pm \frac{ 4 }{ 4 } , ~ \pm \frac{ 4 }{ 5 } , ~ \pm \frac{ 4 }{ 10 } , ~ \pm \frac{ 4 }{ 20 } , ~ \pm \frac{ 5 }{ 1 } , ~ \pm \frac{ 5 }{ 2 } , ~ \pm \frac{ 5 }{ 4 } , ~ \pm \frac{ 5 }{ 5 } , ~ \pm \frac{ 5 }{ 10 } , ~ \pm \frac{ 5 }{ 20 } , ~ \pm \frac{ 6 }{ 1 } , ~ \pm \frac{ 6 }{ 2 } , ~ \pm \frac{ 6 }{ 4 } , ~ \pm \frac{ 6 }{ 5 } , ~ \pm \frac{ 6 }{ 10 } , ~ \pm \frac{ 6 }{ 20 } , ~ \pm \frac{ 8 }{ 1 } , ~ \pm \frac{ 8 }{ 2 } , ~ \pm \frac{ 8 }{ 4 } , ~ \pm \frac{ 8 }{ 5 } , ~ \pm \frac{ 8 }{ 10 } , ~ \pm \frac{ 8 }{ 20 } , ~ \pm \frac{ 9 }{ 1 } , ~ \pm \frac{ 9 }{ 2 } , ~ \pm \frac{ 9 }{ 4 } , ~ \pm \frac{ 9 }{ 5 } , ~ \pm \frac{ 9 }{ 10 } , ~ \pm \frac{ 9 }{ 20 } , ~ \pm \frac{ 10 }{ 1 } , ~ \pm \frac{ 10 }{ 2 } , ~ \pm \frac{ 10 }{ 4 } , ~ \pm \frac{ 10 }{ 5 } , ~ \pm \frac{ 10 }{ 10 } , ~ \pm \frac{ 10 }{ 20 } , ~ \pm \frac{ 12 }{ 1 } , ~ \pm \frac{ 12 }{ 2 } , ~ \pm \frac{ 12 }{ 4 } , ~ \pm \frac{ 12 }{ 5 } , ~ \pm \frac{ 12 }{ 10 } , ~ \pm \frac{ 12 }{ 20 } , ~ \pm \frac{ 15 }{ 1 } , ~ \pm \frac{ 15 }{ 2 } , ~ \pm \frac{ 15 }{ 4 } , ~ \pm \frac{ 15 }{ 5 } , ~ \pm \frac{ 15 }{ 10 } , ~ \pm \frac{ 15 }{ 20 } , ~ \pm \frac{ 16 }{ 1 } , ~ \pm \frac{ 16 }{ 2 } , ~ \pm \frac{ 16 }{ 4 } , ~ \pm \frac{ 16 }{ 5 } , ~ \pm \frac{ 16 }{ 10 } , ~ \pm \frac{ 16 }{ 20 } , ~ \pm \frac{ 18 }{ 1 } , ~ \pm \frac{ 18 }{ 2 } , ~ \pm \frac{ 18 }{ 4 } , ~ \pm \frac{ 18 }{ 5 } , ~ \pm \frac{ 18 }{ 10 } , ~ \pm \frac{ 18 }{ 20 } , ~ \pm \frac{ 20 }{ 1 } , ~ \pm \frac{ 20 }{ 2 } , ~ \pm \frac{ 20 }{ 4 } , ~ \pm \frac{ 20 }{ 5 } , ~ \pm \frac{ 20 }{ 10 } , ~ \pm \frac{ 20 }{ 20 } , ~ \pm \frac{ 24 }{ 1 } , ~ \pm \frac{ 24 }{ 2 } , ~ \pm \frac{ 24 }{ 4 } , ~ \pm \frac{ 24 }{ 5 } , ~ \pm \frac{ 24 }{ 10 } , ~ \pm \frac{ 24 }{ 20 } , ~ \pm \frac{ 30 }{ 1 } , ~ \pm \frac{ 30 }{ 2 } , ~ \pm \frac{ 30 }{ 4 } , ~ \pm \frac{ 30 }{ 5 } , ~ \pm \frac{ 30 }{ 10 } , ~ \pm \frac{ 30 }{ 20 } , ~ \pm \frac{ 32 }{ 1 } , ~ \pm \frac{ 32 }{ 2 } , ~ \pm \frac{ 32 }{ 4 } , ~ \pm \frac{ 32 }{ 5 } , ~ \pm \frac{ 32 }{ 10 } , ~ \pm \frac{ 32 }{ 20 } , ~ \pm \frac{ 36 }{ 1 } , ~ \pm \frac{ 36 }{ 2 } , ~ \pm \frac{ 36 }{ 4 } , ~ \pm \frac{ 36 }{ 5 } , ~ \pm \frac{ 36 }{ 10 } , ~ \pm \frac{ 36 }{ 20 } , ~ \pm \frac{ 40 }{ 1 } , ~ \pm \frac{ 40 }{ 2 } , ~ \pm \frac{ 40 }{ 4 } , ~ \pm \frac{ 40 }{ 5 } , ~ \pm \frac{ 40 }{ 10 } , ~ \pm \frac{ 40 }{ 20 } , ~ \pm \frac{ 45 }{ 1 } , ~ \pm \frac{ 45 }{ 2 } , ~ \pm \frac{ 45 }{ 4 } , ~ \pm \frac{ 45 }{ 5 } , ~ \pm \frac{ 45 }{ 10 } , ~ \pm \frac{ 45 }{ 20 } , ~ \pm \frac{ 48 }{ 1 } , ~ \pm \frac{ 48 }{ 2 } , ~ \pm \frac{ 48 }{ 4 } , ~ \pm \frac{ 48 }{ 5 } , ~ \pm \frac{ 48 }{ 10 } , ~ \pm \frac{ 48 }{ 20 } , ~ \pm \frac{ 60 }{ 1 } , ~ \pm \frac{ 60 }{ 2 } , ~ \pm \frac{ 60 }{ 4 } , ~ \pm \frac{ 60 }{ 5 } , ~ \pm \frac{ 60 }{ 10 } , ~ \pm \frac{ 60 }{ 20 } , ~ \pm \frac{ 64 }{ 1 } , ~ \pm \frac{ 64 }{ 2 } , ~ \pm \frac{ 64 }{ 4 } , ~ \pm \frac{ 64 }{ 5 } , ~ \pm \frac{ 64 }{ 10 } , ~ \pm \frac{ 64 }{ 20 } , ~ \pm \frac{ 72 }{ 1 } , ~ \pm \frac{ 72 }{ 2 } , ~ \pm \frac{ 72 }{ 4 } , ~ \pm \frac{ 72 }{ 5 } , ~ \pm \frac{ 72 }{ 10 } , ~ \pm \frac{ 72 }{ 20 } , ~ \pm \frac{ 80 }{ 1 } , ~ \pm \frac{ 80 }{ 2 } , ~ \pm \frac{ 80 }{ 4 } , ~ \pm \frac{ 80 }{ 5 } , ~ \pm \frac{ 80 }{ 10 } , ~ \pm \frac{ 80 }{ 20 } , ~ \pm \frac{ 90 }{ 1 } , ~ \pm \frac{ 90 }{ 2 } , ~ \pm \frac{ 90 }{ 4 } , ~ \pm \frac{ 90 }{ 5 } , ~ \pm \frac{ 90 }{ 10 } , ~ \pm \frac{ 90 }{ 20 } , ~ \pm \frac{ 96 }{ 1 } , ~ \pm \frac{ 96 }{ 2 } , ~ \pm \frac{ 96 }{ 4 } , ~ \pm \frac{ 96 }{ 5 } , ~ \pm \frac{ 96 }{ 10 } , ~ \pm \frac{ 96 }{ 20 } , ~ \pm \frac{ 120 }{ 1 } , ~ \pm \frac{ 120 }{ 2 } , ~ \pm \frac{ 120 }{ 4 } , ~ \pm \frac{ 120 }{ 5 } , ~ \pm \frac{ 120 }{ 10 } , ~ \pm \frac{ 120 }{ 20 } , ~ \pm \frac{ 128 }{ 1 } , ~ \pm \frac{ 128 }{ 2 } , ~ \pm \frac{ 128 }{ 4 } , ~ \pm \frac{ 128 }{ 5 } , ~ \pm \frac{ 128 }{ 10 } , ~ \pm \frac{ 128 }{ 20 } , ~ \pm \frac{ 144 }{ 1 } , ~ \pm \frac{ 144 }{ 2 } , ~ \pm \frac{ 144 }{ 4 } , ~ \pm \frac{ 144 }{ 5 } , ~ \pm \frac{ 144 }{ 10 } , ~ \pm \frac{ 144 }{ 20 } , ~ \pm \frac{ 160 }{ 1 } , ~ \pm \frac{ 160 }{ 2 } , ~ \pm \frac{ 160 }{ 4 } , ~ \pm \frac{ 160 }{ 5 } , ~ \pm \frac{ 160 }{ 10 } , ~ \pm \frac{ 160 }{ 20 } , ~ \pm \frac{ 180 }{ 1 } , ~ \pm \frac{ 180 }{ 2 } , ~ \pm \frac{ 180 }{ 4 } , ~ \pm \frac{ 180 }{ 5 } , ~ \pm \frac{ 180 }{ 10 } , ~ \pm \frac{ 180 }{ 20 } , ~ \pm \frac{ 192 }{ 1 } , ~ \pm \frac{ 192 }{ 2 } , ~ \pm \frac{ 192 }{ 4 } , ~ \pm \frac{ 192 }{ 5 } , ~ \pm \frac{ 192 }{ 10 } , ~ \pm \frac{ 192 }{ 20 } , ~ \pm \frac{ 240 }{ 1 } , ~ \pm \frac{ 240 }{ 2 } , ~ \pm \frac{ 240 }{ 4 } , ~ \pm \frac{ 240 }{ 5 } , ~ \pm \frac{ 240 }{ 10 } , ~ \pm \frac{ 240 }{ 20 } , ~ \pm \frac{ 288 }{ 1 } , ~ \pm \frac{ 288 }{ 2 } , ~ \pm \frac{ 288 }{ 4 } , ~ \pm \frac{ 288 }{ 5 } , ~ \pm \frac{ 288 }{ 10 } , ~ \pm \frac{ 288 }{ 20 } , ~ \pm \frac{ 320 }{ 1 } , ~ \pm \frac{ 320 }{ 2 } , ~ \pm \frac{ 320 }{ 4 } , ~ \pm \frac{ 320 }{ 5 } , ~ \pm \frac{ 320 }{ 10 } , ~ \pm \frac{ 320 }{ 20 } , ~ \pm \frac{ 360 }{ 1 } , ~ \pm \frac{ 360 }{ 2 } , ~ \pm \frac{ 360 }{ 4 } , ~ \pm \frac{ 360 }{ 5 } , ~ \pm \frac{ 360 }{ 10 } , ~ \pm \frac{ 360 }{ 20 } , ~ \pm \frac{ 384 }{ 1 } , ~ \pm \frac{ 384 }{ 2 } , ~ \pm \frac{ 384 }{ 4 } , ~ \pm \frac{ 384 }{ 5 } , ~ \pm \frac{ 384 }{ 10 } , ~ \pm \frac{ 384 }{ 20 } , ~ \pm \frac{ 480 }{ 1 } , ~ \pm \frac{ 480 }{ 2 } , ~ \pm \frac{ 480 }{ 4 } , ~ \pm \frac{ 480 }{ 5 } , ~ \pm \frac{ 480 }{ 10 } , ~ \pm \frac{ 480 }{ 20 } , ~ \pm \frac{ 576 }{ 1 } , ~ \pm \frac{ 576 }{ 2 } , ~ \pm \frac{ 576 }{ 4 } , ~ \pm \frac{ 576 }{ 5 } , ~ \pm \frac{ 576 }{ 10 } , ~ \pm \frac{ 576 }{ 20 } , ~ \pm \frac{ 640 }{ 1 } , ~ \pm \frac{ 640 }{ 2 } , ~ \pm \frac{ 640 }{ 4 } , ~ \pm \frac{ 640 }{ 5 } , ~ \pm \frac{ 640 }{ 10 } , ~ \pm \frac{ 640 }{ 20 } , ~ \pm \frac{ 720 }{ 1 } , ~ \pm \frac{ 720 }{ 2 } , ~ \pm \frac{ 720 }{ 4 } , ~ \pm \frac{ 720 }{ 5 } , ~ \pm \frac{ 720 }{ 10 } , ~ \pm \frac{ 720 }{ 20 } , ~ \pm \frac{ 960 }{ 1 } , ~ \pm \frac{ 960 }{ 2 } , ~ \pm \frac{ 960 }{ 4 } , ~ \pm \frac{ 960 }{ 5 } , ~ \pm \frac{ 960 }{ 10 } , ~ \pm \frac{ 960 }{ 20 } , ~ \pm \frac{ 1152 }{ 1 } , ~ \pm \frac{ 1152 }{ 2 } , ~ \pm \frac{ 1152 }{ 4 } , ~ \pm \frac{ 1152 }{ 5 } , ~ \pm \frac{ 1152 }{ 10 } , ~ \pm \frac{ 1152 }{ 20 } , ~ \pm \frac{ 1440 }{ 1 } , ~ \pm \frac{ 1440 }{ 2 } , ~ \pm \frac{ 1440 }{ 4 } , ~ \pm \frac{ 1440 }{ 5 } , ~ \pm \frac{ 1440 }{ 10 } , ~ \pm \frac{ 1440 }{ 20 } , ~ \pm \frac{ 1920 }{ 1 } , ~ \pm \frac{ 1920 }{ 2 } , ~ \pm \frac{ 1920 }{ 4 } , ~ \pm \frac{ 1920 }{ 5 } , ~ \pm \frac{ 1920 }{ 10 } , ~ \pm \frac{ 1920 }{ 20 } , ~ \pm \frac{ 2880 }{ 1 } , ~ \pm \frac{ 2880 }{ 2 } , ~ \pm \frac{ 2880 }{ 4 } , ~ \pm \frac{ 2880 }{ 5 } , ~ \pm \frac{ 2880 }{ 10 } , ~ \pm \frac{ 2880 }{ 20 } , ~ \pm \frac{ 5760 }{ 1 } , ~ \pm \frac{ 5760 }{ 2 } , ~ \pm \frac{ 5760 }{ 4 } , ~ \pm \frac{ 5760 }{ 5 } , ~ \pm \frac{ 5760 }{ 10 } , ~ \pm \frac{ 5760 }{ 20 } ~ $$

Substitute the POSSIBLE roots one by one into the polynomial to find the actual roots. Start first with the whole numbers.

If we plug these values into the polynomial $ P(x) $, we obtain $ P(-2) = 0 $.

To find remaining zeros we use Factor Theorem. This theorem states that if $\frac{p}{q}$ is root of the polynomial then this polynomial can be divided with $ \color{blue}{q x - p} $. In this example:

Divide $ P(x) $ with $ \color{blue}{x + 2} $

$$ \frac{ 20x^{4}+380x^{3}+2480x^{2}+6480x+5760 }{ \color{blue}{ x + 2 } } = 20x^{3}+340x^{2}+1800x+2880 $$

Polynomial $ 20x^{3}+340x^{2}+1800x+2880 $ can be used to find the remaining roots.

Use the same procedure to find roots of $ 20x^{3}+340x^{2}+1800x+2880 $

When you get second degree polynomial use step-by-step quadratic equation solver to find two remaining roots.

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