$$-(-2x^4-3x^3+4x^2)(2x^2+4x) = 0$$

$$ \begin{matrix}x_1 = 0 & x_2 = -2 & x_3 = -\dfrac{ 3 }{ 4 }-\dfrac{\sqrt{ 41 }}{ 4 } \\[1 em] x_4 = -\dfrac{ 3 }{ 4 }+\dfrac{\sqrt{ 41 }}{ 4 } & \\[1 em] \end{matrix} $$

In order to solve $ \color{blue}{ 4x^{6}+14x^{5}+4x^{4}-16x^{3} = 0 } $, first we need to factor our $ x^3 $.

$$ 4x^{6}+14x^{5}+4x^{4}-16x^{3} = x^3 \left( 4x^{3}+14x^{2}+4x-16 \right) $$

$ x = 0 $ is a root of multiplicity $ 3 $.

The remaining roots can be found by solving equation $ 4x^{3}+14x^{2}+4x-16 = 0$.

$ \color{blue}{ 4x^{3}+14x^{2}+4x-16 } $ is a polynomial of degree 3. 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 ( 4 ) are 1 2 4 .The factors of the constant term (-16) are 1 2 4 8 16 . Then the Rational Roots Tests yields the following possible solutions:

$$ \pm \frac{ 1 }{ 1 } , ~ \pm \frac{ 1 }{ 2 } , ~ \pm \frac{ 1 }{ 4 } , ~ \pm \frac{ 2 }{ 1 } , ~ \pm \frac{ 2 }{ 2 } , ~ \pm \frac{ 2 }{ 4 } , ~ \pm \frac{ 4 }{ 1 } , ~ \pm \frac{ 4 }{ 2 } , ~ \pm \frac{ 4 }{ 4 } , ~ \pm \frac{ 8 }{ 1 } , ~ \pm \frac{ 8 }{ 2 } , ~ \pm \frac{ 8 }{ 4 } , ~ \pm \frac{ 16 }{ 1 } , ~ \pm \frac{ 16 }{ 2 } , ~ \pm \frac{ 16 }{ 4 } ~ $$

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} $

$$ \frac{ 4x^{3}+14x^{2}+4x-16 }{ \color{blue}{ x + 2 } } = 4x^{2}+6x-8 $$

Polynomial $ 4x^{2}+6x-8 $ can be used to find the remaining roots.

$ \color{blue}{ 4x^{2}+6x-8 } $ is a second degree polynomial. For a detailed answer how to find its roots you can use step-by-step quadratic equation solver.

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