$$\frac{x^3-27}{4x}\frac{2x+4}{x^2-x-6} = 0$$

$$ \begin{matrix}x_1 = 0 & x_2 = -2 & x_3 = 3 \\[1 em] x_4 = -\dfrac{ 3 }{ 2 }+\dfrac{ 3 \sqrt{ 3}}{ 2 }i & x_5 = -\dfrac{ 3 }{ 2 }-\dfrac{ 3 \sqrt{ 3}}{ 2 }i \\[1 em] \end{matrix} $$

In order to solve $ \color{blue}{ 2x^{6}+4x^{5}-54x^{3}-108x^{2} = 0 } $, first we need to factor our $ x^2 $.

$$ 2x^{6}+4x^{5}-54x^{3}-108x^{2} = x^2 \left( 2x^{4}+4x^{3}-54x-108 \right) $$

$ x = 0 $ is a root of multiplicity $ 2 $.

The remaining roots can be found by solving equation $ 2x^{4}+4x^{3}-54x-108 = 0$.

$ \color{blue}{ 2x^{4}+4x^{3}-54x-108 } $ 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 ( 2 ) are 1 2 .The factors of the constant term (-108) are 1 2 3 4 6 9 12 18 27 36 54 108 . Then the Rational Roots Tests yields the following possible solutions:

$$ \pm \frac{ 1 }{ 1 } , ~ \pm \frac{ 1 }{ 2 } , ~ \pm \frac{ 2 }{ 1 } , ~ \pm \frac{ 2 }{ 2 } , ~ \pm \frac{ 3 }{ 1 } , ~ \pm \frac{ 3 }{ 2 } , ~ \pm \frac{ 4 }{ 1 } , ~ \pm \frac{ 4 }{ 2 } , ~ \pm \frac{ 6 }{ 1 } , ~ \pm \frac{ 6 }{ 2 } , ~ \pm \frac{ 9 }{ 1 } , ~ \pm \frac{ 9 }{ 2 } , ~ \pm \frac{ 12 }{ 1 } , ~ \pm \frac{ 12 }{ 2 } , ~ \pm \frac{ 18 }{ 1 } , ~ \pm \frac{ 18 }{ 2 } , ~ \pm \frac{ 27 }{ 1 } , ~ \pm \frac{ 27 }{ 2 } , ~ \pm \frac{ 36 }{ 1 } , ~ \pm \frac{ 36 }{ 2 } , ~ \pm \frac{ 54 }{ 1 } , ~ \pm \frac{ 54 }{ 2 } , ~ \pm \frac{ 108 }{ 1 } , ~ \pm \frac{ 108 }{ 2 } ~ $$

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} $

$$ \frac{ 2x^{4}+4x^{3}-54x-108 }{ \color{blue}{ x + 2 } } = 2x^{3}-54 $$

Polynomial $ 2x^{3}-54 $ can be used to find the remaining roots.

Use the same procedure to find roots of $ 2x^{3}-54 $

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

Polynomial Equations Solver