$$-x^3+30x^2+600x-8000 = 0$$
$$ \begin{matrix}x_1 = 10 & x_2 = -20 & x_3 = 40 \end{matrix} $$
$ \color{blue}{ -x^{3}+30x^{2}+600x-8000 } $ 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 (-8000) are 1 2 4 5 8 10 16 20 25 32 40 50 64 80 100 125 160 200 250 320 400 500 800 1000 1600 2000 4000 8000 . Then the Rational Roots Tests yields the following possible solutions:
$$ \pm \frac{ 1 }{ 1 } , ~ \pm \frac{ 2 }{ 1 } , ~ \pm \frac{ 4 }{ 1 } , ~ \pm \frac{ 5 }{ 1 } , ~ \pm \frac{ 8 }{ 1 } , ~ \pm \frac{ 10 }{ 1 } , ~ \pm \frac{ 16 }{ 1 } , ~ \pm \frac{ 20 }{ 1 } , ~ \pm \frac{ 25 }{ 1 } , ~ \pm \frac{ 32 }{ 1 } , ~ \pm \frac{ 40 }{ 1 } , ~ \pm \frac{ 50 }{ 1 } , ~ \pm \frac{ 64 }{ 1 } , ~ \pm \frac{ 80 }{ 1 } , ~ \pm \frac{ 100 }{ 1 } , ~ \pm \frac{ 125 }{ 1 } , ~ \pm \frac{ 160 }{ 1 } , ~ \pm \frac{ 200 }{ 1 } , ~ \pm \frac{ 250 }{ 1 } , ~ \pm \frac{ 320 }{ 1 } , ~ \pm \frac{ 400 }{ 1 } , ~ \pm \frac{ 500 }{ 1 } , ~ \pm \frac{ 800 }{ 1 } , ~ \pm \frac{ 1000 }{ 1 } , ~ \pm \frac{ 1600 }{ 1 } , ~ \pm \frac{ 2000 }{ 1 } , ~ \pm \frac{ 4000 }{ 1 } , ~ \pm \frac{ 8000 }{ 1 } ~ $$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(10) = 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 - 10} $
$$ \frac{ -x^{3}+30x^{2}+600x-8000 }{ \color{blue}{ x - 10 } } = -x^{2}+20x+800 $$Polynomial $ -x^{2}+20x+800 $ can be used to find the remaining roots.
$ \color{blue}{ -x^{2}+20x+800 } $ is a second degree polynomial. For a detailed answer how to find its roots you can use step-by-step quadratic equation solver.