$$3(x+4)\cdot3(x+4)\cdot3(x+4) = 0$$
$$ x = -4 $$
$ \color{blue}{ 27x^{3}+324x^{2}+1296x+1728 } $ 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 ( 27 ) are 1 3 9 27 .The factors of the constant term (1728) are 1 2 3 4 6 8 9 12 16 18 24 27 32 36 48 54 64 72 96 108 144 192 216 288 432 576 864 1728 . Then the Rational Roots Tests yields the following possible solutions:
$$ \pm \frac{ 1 }{ 1 } , ~ \pm \frac{ 1 }{ 3 } , ~ \pm \frac{ 1 }{ 9 } , ~ \pm \frac{ 1 }{ 27 } , ~ \pm \frac{ 2 }{ 1 } , ~ \pm \frac{ 2 }{ 3 } , ~ \pm \frac{ 2 }{ 9 } , ~ \pm \frac{ 2 }{ 27 } , ~ \pm \frac{ 3 }{ 1 } , ~ \pm \frac{ 3 }{ 3 } , ~ \pm \frac{ 3 }{ 9 } , ~ \pm \frac{ 3 }{ 27 } , ~ \pm \frac{ 4 }{ 1 } , ~ \pm \frac{ 4 }{ 3 } , ~ \pm \frac{ 4 }{ 9 } , ~ \pm \frac{ 4 }{ 27 } , ~ \pm \frac{ 6 }{ 1 } , ~ \pm \frac{ 6 }{ 3 } , ~ \pm \frac{ 6 }{ 9 } , ~ \pm \frac{ 6 }{ 27 } , ~ \pm \frac{ 8 }{ 1 } , ~ \pm \frac{ 8 }{ 3 } , ~ \pm \frac{ 8 }{ 9 } , ~ \pm \frac{ 8 }{ 27 } , ~ \pm \frac{ 9 }{ 1 } , ~ \pm \frac{ 9 }{ 3 } , ~ \pm \frac{ 9 }{ 9 } , ~ \pm \frac{ 9 }{ 27 } , ~ \pm \frac{ 12 }{ 1 } , ~ \pm \frac{ 12 }{ 3 } , ~ \pm \frac{ 12 }{ 9 } , ~ \pm \frac{ 12 }{ 27 } , ~ \pm \frac{ 16 }{ 1 } , ~ \pm \frac{ 16 }{ 3 } , ~ \pm \frac{ 16 }{ 9 } , ~ \pm \frac{ 16 }{ 27 } , ~ \pm \frac{ 18 }{ 1 } , ~ \pm \frac{ 18 }{ 3 } , ~ \pm \frac{ 18 }{ 9 } , ~ \pm \frac{ 18 }{ 27 } , ~ \pm \frac{ 24 }{ 1 } , ~ \pm \frac{ 24 }{ 3 } , ~ \pm \frac{ 24 }{ 9 } , ~ \pm \frac{ 24 }{ 27 } , ~ \pm \frac{ 27 }{ 1 } , ~ \pm \frac{ 27 }{ 3 } , ~ \pm \frac{ 27 }{ 9 } , ~ \pm \frac{ 27 }{ 27 } , ~ \pm \frac{ 32 }{ 1 } , ~ \pm \frac{ 32 }{ 3 } , ~ \pm \frac{ 32 }{ 9 } , ~ \pm \frac{ 32 }{ 27 } , ~ \pm \frac{ 36 }{ 1 } , ~ \pm \frac{ 36 }{ 3 } , ~ \pm \frac{ 36 }{ 9 } , ~ \pm \frac{ 36 }{ 27 } , ~ \pm \frac{ 48 }{ 1 } , ~ \pm \frac{ 48 }{ 3 } , ~ \pm \frac{ 48 }{ 9 } , ~ \pm \frac{ 48 }{ 27 } , ~ \pm \frac{ 54 }{ 1 } , ~ \pm \frac{ 54 }{ 3 } , ~ \pm \frac{ 54 }{ 9 } , ~ \pm \frac{ 54 }{ 27 } , ~ \pm \frac{ 64 }{ 1 } , ~ \pm \frac{ 64 }{ 3 } , ~ \pm \frac{ 64 }{ 9 } , ~ \pm \frac{ 64 }{ 27 } , ~ \pm \frac{ 72 }{ 1 } , ~ \pm \frac{ 72 }{ 3 } , ~ \pm \frac{ 72 }{ 9 } , ~ \pm \frac{ 72 }{ 27 } , ~ \pm \frac{ 96 }{ 1 } , ~ \pm \frac{ 96 }{ 3 } , ~ \pm \frac{ 96 }{ 9 } , ~ \pm \frac{ 96 }{ 27 } , ~ \pm \frac{ 108 }{ 1 } , ~ \pm \frac{ 108 }{ 3 } , ~ \pm \frac{ 108 }{ 9 } , ~ \pm \frac{ 108 }{ 27 } , ~ \pm \frac{ 144 }{ 1 } , ~ \pm \frac{ 144 }{ 3 } , ~ \pm \frac{ 144 }{ 9 } , ~ \pm \frac{ 144 }{ 27 } , ~ \pm \frac{ 192 }{ 1 } , ~ \pm \frac{ 192 }{ 3 } , ~ \pm \frac{ 192 }{ 9 } , ~ \pm \frac{ 192 }{ 27 } , ~ \pm \frac{ 216 }{ 1 } , ~ \pm \frac{ 216 }{ 3 } , ~ \pm \frac{ 216 }{ 9 } , ~ \pm \frac{ 216 }{ 27 } , ~ \pm \frac{ 288 }{ 1 } , ~ \pm \frac{ 288 }{ 3 } , ~ \pm \frac{ 288 }{ 9 } , ~ \pm \frac{ 288 }{ 27 } , ~ \pm \frac{ 432 }{ 1 } , ~ \pm \frac{ 432 }{ 3 } , ~ \pm \frac{ 432 }{ 9 } , ~ \pm \frac{ 432 }{ 27 } , ~ \pm \frac{ 576 }{ 1 } , ~ \pm \frac{ 576 }{ 3 } , ~ \pm \frac{ 576 }{ 9 } , ~ \pm \frac{ 576 }{ 27 } , ~ \pm \frac{ 864 }{ 1 } , ~ \pm \frac{ 864 }{ 3 } , ~ \pm \frac{ 864 }{ 9 } , ~ \pm \frac{ 864 }{ 27 } , ~ \pm \frac{ 1728 }{ 1 } , ~ \pm \frac{ 1728 }{ 3 } , ~ \pm \frac{ 1728 }{ 9 } , ~ \pm \frac{ 1728 }{ 27 } ~ $$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(-4) = 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 + 4} $
$$ \frac{ 27x^{3}+324x^{2}+1296x+1728 }{ \color{blue}{ x + 4 } } = 27x^{2}+216x+432 $$Polynomial $ 27x^{2}+216x+432 $ can be used to find the remaining roots.
$ \color{blue}{ 27x^{2}+216x+432 } $ is a second degree polynomial. For a detailed answer how to find its roots you can use step-by-step quadratic equation solver.