$$x^3+44x^2+588x-3120 = 0$$
$$ \begin{matrix}x_1 = 4 & x_2 = -24+2 \sqrt{ 51 }i & x_3 = -24-2 \sqrt{ 51 }i \end{matrix} $$
$ \color{blue}{ x^{3}+44x^{2}+588x-3120 } $ 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 factor of the leading coefficient ( 1 ) is 1 .The factors of the constant term (-3120) are 1 2 3 4 5 6 8 10 12 13 15 16 20 24 26 30 39 40 48 52 60 65 78 80 104 120 130 156 195 208 240 260 312 390 520 624 780 1040 1560 3120 . Then the Rational Roots Tests yields the following possible solutions:
$$ \pm \frac{ 1 }{ 1 } , ~ \pm \frac{ 2 }{ 1 } , ~ \pm \frac{ 3 }{ 1 } , ~ \pm \frac{ 4 }{ 1 } , ~ \pm \frac{ 5 }{ 1 } , ~ \pm \frac{ 6 }{ 1 } , ~ \pm \frac{ 8 }{ 1 } , ~ \pm \frac{ 10 }{ 1 } , ~ \pm \frac{ 12 }{ 1 } , ~ \pm \frac{ 13 }{ 1 } , ~ \pm \frac{ 15 }{ 1 } , ~ \pm \frac{ 16 }{ 1 } , ~ \pm \frac{ 20 }{ 1 } , ~ \pm \frac{ 24 }{ 1 } , ~ \pm \frac{ 26 }{ 1 } , ~ \pm \frac{ 30 }{ 1 } , ~ \pm \frac{ 39 }{ 1 } , ~ \pm \frac{ 40 }{ 1 } , ~ \pm \frac{ 48 }{ 1 } , ~ \pm \frac{ 52 }{ 1 } , ~ \pm \frac{ 60 }{ 1 } , ~ \pm \frac{ 65 }{ 1 } , ~ \pm \frac{ 78 }{ 1 } , ~ \pm \frac{ 80 }{ 1 } , ~ \pm \frac{ 104 }{ 1 } , ~ \pm \frac{ 120 }{ 1 } , ~ \pm \frac{ 130 }{ 1 } , ~ \pm \frac{ 156 }{ 1 } , ~ \pm \frac{ 195 }{ 1 } , ~ \pm \frac{ 208 }{ 1 } , ~ \pm \frac{ 240 }{ 1 } , ~ \pm \frac{ 260 }{ 1 } , ~ \pm \frac{ 312 }{ 1 } , ~ \pm \frac{ 390 }{ 1 } , ~ \pm \frac{ 520 }{ 1 } , ~ \pm \frac{ 624 }{ 1 } , ~ \pm \frac{ 780 }{ 1 } , ~ \pm \frac{ 1040 }{ 1 } , ~ \pm \frac{ 1560 }{ 1 } , ~ \pm \frac{ 3120 }{ 1 } ~ $$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(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} $
$$ \frac{ x^{3}+44x^{2}+588x-3120 }{ \color{blue}{ x - 4 } } = x^{2}+48x+780 $$Polynomial $ x^{2}+48x+780 $ can be used to find the remaining roots.
$ \color{blue}{ x^{2}+48x+780 } $ is a second degree polynomial. For a detailed answer how to find its roots you can use step-by-step quadratic equation solver.