$$x^4-34x^3-80x^2+2720x = 0$$
$$ \begin{matrix}x_1 = 0 & x_2 = 34 & x_3 = -4 \sqrt{ 5 } \\[1 em] x_4 = 4 \sqrt{ 5 } & \\[1 em] \end{matrix} $$
In order to solve $ \color{blue}{ x^{4}-34x^{3}-80x^{2}+2720x = 0 } $, first we need to factor our $ x $.
$$ x^{4}-34x^{3}-80x^{2}+2720x = x \left( x^{3}-34x^{2}-80x+2720 \right) $$$ x = 0 $ is a root of multiplicity $ 1 $.
The remaining roots can be found by solving equation $ x^{3}-34x^{2}-80x+2720 = 0$.
$ \color{blue}{ x^{3}-34x^{2}-80x+2720 } $ 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 (2720) are 1 2 4 5 8 10 16 17 20 32 34 40 68 80 85 136 160 170 272 340 544 680 1360 2720 . Then the Rational Roots Tests yields the following possible solutions:
$$ \pm \frac{ 1 }{ 1 } , ~ \pm \frac{ 2 }{ 1 } , ~ \pm \frac{ 4 }{ 1 } , ~ \pm \frac{ 5 }{ 1 } , ~ \pm \frac{ 8 }{ 1 } , ~ \pm \frac{ 10 }{ 1 } , ~ \pm \frac{ 16 }{ 1 } , ~ \pm \frac{ 17 }{ 1 } , ~ \pm \frac{ 20 }{ 1 } , ~ \pm \frac{ 32 }{ 1 } , ~ \pm \frac{ 34 }{ 1 } , ~ \pm \frac{ 40 }{ 1 } , ~ \pm \frac{ 68 }{ 1 } , ~ \pm \frac{ 80 }{ 1 } , ~ \pm \frac{ 85 }{ 1 } , ~ \pm \frac{ 136 }{ 1 } , ~ \pm \frac{ 160 }{ 1 } , ~ \pm \frac{ 170 }{ 1 } , ~ \pm \frac{ 272 }{ 1 } , ~ \pm \frac{ 340 }{ 1 } , ~ \pm \frac{ 544 }{ 1 } , ~ \pm \frac{ 680 }{ 1 } , ~ \pm \frac{ 1360 }{ 1 } , ~ \pm \frac{ 2720 }{ 1 } ~ $$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(34) = 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 - 34} $
$$ \frac{ x^{3}-34x^{2}-80x+2720 }{ \color{blue}{ x - 34 } } = x^{2}-80 $$Polynomial $ x^{2}-80 $ can be used to find the remaining roots.
$ \color{blue}{ x^{2}-80 } $ is a second degree polynomial. For a detailed answer how to find its roots you can use step-by-step quadratic equation solver.