$$x(x-3)^4(x+6)^2 = 0$$
$$ \begin{matrix}x_1 = 0 & x_2 = 3 & x_3 = -6 \end{matrix} $$
In order to solve $ \color{blue}{ x^{7}-54x^{5}+108x^{4}+729x^{3}-2916x^{2}+2916x = 0 } $, first we need to factor our $ x $.
$$ x^{7}-54x^{5}+108x^{4}+729x^{3}-2916x^{2}+2916x = x \left( x^{6}-54x^{4}+108x^{3}+729x^{2}-2916x+2916 \right) $$$ x = 0 $ is a root of multiplicity $ 1 $.
The remaining roots can be found by solving equation $ x^{6}-54x^{4}+108x^{3}+729x^{2}-2916x+2916 = 0$.
$ \color{blue}{ x^{6}-54x^{4}+108x^{3}+729x^{2}-2916x+2916 } $ is a polynomial of degree 6. 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 factor of the leading coefficient ( 1 ) is 1 .The factors of the constant term (2916) are 1 2 3 4 6 9 12 18 27 36 54 81 108 162 243 324 486 729 972 1458 2916 . Then the Rational Roots Tests yields the following possible solutions:
$$ \pm \frac{ 1 }{ 1 } , ~ \pm \frac{ 2 }{ 1 } , ~ \pm \frac{ 3 }{ 1 } , ~ \pm \frac{ 4 }{ 1 } , ~ \pm \frac{ 6 }{ 1 } , ~ \pm \frac{ 9 }{ 1 } , ~ \pm \frac{ 12 }{ 1 } , ~ \pm \frac{ 18 }{ 1 } , ~ \pm \frac{ 27 }{ 1 } , ~ \pm \frac{ 36 }{ 1 } , ~ \pm \frac{ 54 }{ 1 } , ~ \pm \frac{ 81 }{ 1 } , ~ \pm \frac{ 108 }{ 1 } , ~ \pm \frac{ 162 }{ 1 } , ~ \pm \frac{ 243 }{ 1 } , ~ \pm \frac{ 324 }{ 1 } , ~ \pm \frac{ 486 }{ 1 } , ~ \pm \frac{ 729 }{ 1 } , ~ \pm \frac{ 972 }{ 1 } , ~ \pm \frac{ 1458 }{ 1 } , ~ \pm \frac{ 2916 }{ 1 } ~ $$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(3) = 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 - 3} $
$$ \frac{ x^{6}-54x^{4}+108x^{3}+729x^{2}-2916x+2916 }{ \color{blue}{ x - 3 } } = x^{5}+3x^{4}-45x^{3}-27x^{2}+648x-972 $$Polynomial $ x^{5}+3x^{4}-45x^{3}-27x^{2}+648x-972 $ can be used to find the remaining roots.
Use the same procedure to find roots of $ x^{5}+3x^{4}-45x^{3}-27x^{2}+648x-972 $
When you get second degree polynomial use step-by-step quadratic equation solver to find two remaining roots.