$ \color{blue}{ x^{4}-236x^{3}+17780x^{2}-465088x+2699520 } $ is a polynomial of degree 4. 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 (2699520) are 1 2 3 4 5 6 8 10 12 15 16 19 20 24 30 32 37 38 40 48 57 60 64 74 76 80 95 96 111 114 120 128 148 152 160 185 190 192 222 228 240 256 285 296 304 320 370 380 384 444 456 480 555 570 592 608 640 703 740 760 768 888 912 960 1110 1140 1184 1216 1280 1406 1480 1520 1776 1824 1920 2109 2220 2280 2368 2432 2812 2960 3040 3515 3552 3648 3840 4218 4440 4560 4736 4864 5624 5920 6080 7030 7104 7296 8436 8880 9120 9472 10545 11248 11840 12160 14060 14208 14592 16872 17760 18240 21090 22496 23680 24320 28120 28416 33744 35520 36480 42180 44992 47360 56240 67488 71040 72960 84360 89984 112480 134976 142080 168720 179968 224960 269952 337440 449920 539904 674880 899840 1349760 2699520 . 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{ 15 }{ 1 } , ~ \pm \frac{ 16 }{ 1 } , ~ \pm \frac{ 19 }{ 1 } , ~ \pm \frac{ 20 }{ 1 } , ~ \pm \frac{ 24 }{ 1 } , ~ \pm \frac{ 30 }{ 1 } , ~ \pm \frac{ 32 }{ 1 } , ~ \pm \frac{ 37 }{ 1 } , ~ \pm \frac{ 38 }{ 1 } , ~ \pm \frac{ 40 }{ 1 } , ~ \pm \frac{ 48 }{ 1 } , ~ \pm \frac{ 57 }{ 1 } , ~ \pm \frac{ 60 }{ 1 } , ~ \pm \frac{ 64 }{ 1 } , ~ \pm \frac{ 74 }{ 1 } , ~ \pm \frac{ 76 }{ 1 } , ~ \pm \frac{ 80 }{ 1 } , ~ \pm \frac{ 95 }{ 1 } , ~ \pm \frac{ 96 }{ 1 } , ~ \pm \frac{ 111 }{ 1 } , ~ \pm \frac{ 114 }{ 1 } , ~ \pm \frac{ 120 }{ 1 } , ~ \pm \frac{ 128 }{ 1 } , ~ \pm \frac{ 148 }{ 1 } , ~ \pm \frac{ 152 }{ 1 } , ~ \pm \frac{ 160 }{ 1 } , ~ \pm \frac{ 185 }{ 1 } , ~ \pm \frac{ 190 }{ 1 } , ~ \pm \frac{ 192 }{ 1 } , ~ \pm \frac{ 222 }{ 1 } , ~ \pm \frac{ 228 }{ 1 } , ~ \pm \frac{ 240 }{ 1 } , ~ \pm \frac{ 256 }{ 1 } , ~ \pm \frac{ 285 }{ 1 } , ~ \pm \frac{ 296 }{ 1 } , ~ \pm \frac{ 304 }{ 1 } , ~ \pm \frac{ 320 }{ 1 } , ~ \pm \frac{ 370 }{ 1 } , ~ \pm \frac{ 380 }{ 1 } , ~ \pm \frac{ 384 }{ 1 } , ~ \pm \frac{ 444 }{ 1 } , ~ \pm \frac{ 456 }{ 1 } , ~ \pm \frac{ 480 }{ 1 } , ~ \pm \frac{ 555 }{ 1 } , ~ \pm \frac{ 570 }{ 1 } , ~ \pm \frac{ 592 }{ 1 } , ~ \pm \frac{ 608 }{ 1 } , ~ \pm \frac{ 640 }{ 1 } , ~ \pm \frac{ 703 }{ 1 } , ~ \pm \frac{ 740 }{ 1 } , ~ \pm \frac{ 760 }{ 1 } , ~ \pm \frac{ 768 }{ 1 } , ~ \pm \frac{ 888 }{ 1 } , ~ \pm \frac{ 912 }{ 1 } , ~ \pm \frac{ 960 }{ 1 } , ~ \pm \frac{ 1110 }{ 1 } , ~ \pm \frac{ 1140 }{ 1 } , ~ \pm \frac{ 1184 }{ 1 } , ~ \pm \frac{ 1216 }{ 1 } , ~ \pm \frac{ 1280 }{ 1 } , ~ \pm \frac{ 1406 }{ 1 } , ~ \pm \frac{ 1480 }{ 1 } , ~ \pm \frac{ 1520 }{ 1 } , ~ \pm \frac{ 1776 }{ 1 } , ~ \pm \frac{ 1824 }{ 1 } , ~ \pm \frac{ 1920 }{ 1 } , ~ \pm \frac{ 2109 }{ 1 } , ~ \pm \frac{ 2220 }{ 1 } , ~ \pm \frac{ 2280 }{ 1 } , ~ \pm \frac{ 2368 }{ 1 } , ~ \pm \frac{ 2432 }{ 1 } , ~ \pm \frac{ 2812 }{ 1 } , ~ \pm \frac{ 2960 }{ 1 } , ~ \pm \frac{ 3040 }{ 1 } , ~ \pm \frac{ 3515 }{ 1 } , ~ \pm \frac{ 3552 }{ 1 } , ~ \pm \frac{ 3648 }{ 1 } , ~ \pm \frac{ 3840 }{ 1 } , ~ \pm \frac{ 4218 }{ 1 } , ~ \pm \frac{ 4440 }{ 1 } , ~ \pm \frac{ 4560 }{ 1 } , ~ \pm \frac{ 4736 }{ 1 } , ~ \pm \frac{ 4864 }{ 1 } , ~ \pm \frac{ 5624 }{ 1 } , ~ \pm \frac{ 5920 }{ 1 } , ~ \pm \frac{ 6080 }{ 1 } , ~ \pm \frac{ 7030 }{ 1 } , ~ \pm \frac{ 7104 }{ 1 } , ~ \pm \frac{ 7296 }{ 1 } , ~ \pm \frac{ 8436 }{ 1 } , ~ \pm \frac{ 8880 }{ 1 } , ~ \pm \frac{ 9120 }{ 1 } , ~ \pm \frac{ 9472 }{ 1 } , ~ \pm \frac{ 10545 }{ 1 } , ~ \pm \frac{ 11248 }{ 1 } , ~ \pm \frac{ 11840 }{ 1 } , ~ \pm \frac{ 12160 }{ 1 } , ~ \pm \frac{ 14060 }{ 1 } , ~ \pm \frac{ 14208 }{ 1 } , ~ \pm \frac{ 14592 }{ 1 } , ~ \pm \frac{ 16872 }{ 1 } , ~ \pm \frac{ 17760 }{ 1 } , ~ \pm \frac{ 18240 }{ 1 } , ~ \pm \frac{ 21090 }{ 1 } , ~ \pm \frac{ 22496 }{ 1 } , ~ \pm \frac{ 23680 }{ 1 } , ~ \pm \frac{ 24320 }{ 1 } , ~ \pm \frac{ 28120 }{ 1 } , ~ \pm \frac{ 28416 }{ 1 } , ~ \pm \frac{ 33744 }{ 1 } , ~ \pm \frac{ 35520 }{ 1 } , ~ \pm \frac{ 36480 }{ 1 } , ~ \pm \frac{ 42180 }{ 1 } , ~ \pm \frac{ 44992 }{ 1 } , ~ \pm \frac{ 47360 }{ 1 } , ~ \pm \frac{ 56240 }{ 1 } , ~ \pm \frac{ 67488 }{ 1 } , ~ \pm \frac{ 71040 }{ 1 } , ~ \pm \frac{ 72960 }{ 1 } , ~ \pm \frac{ 84360 }{ 1 } , ~ \pm \frac{ 89984 }{ 1 } , ~ \pm \frac{ 112480 }{ 1 } , ~ \pm \frac{ 134976 }{ 1 } , ~ \pm \frac{ 142080 }{ 1 } , ~ \pm \frac{ 168720 }{ 1 } , ~ \pm \frac{ 179968 }{ 1 } , ~ \pm \frac{ 224960 }{ 1 } , ~ \pm \frac{ 269952 }{ 1 } , ~ \pm \frac{ 337440 }{ 1 } , ~ \pm \frac{ 449920 }{ 1 } , ~ \pm \frac{ 539904 }{ 1 } , ~ \pm \frac{ 674880 }{ 1 } , ~ \pm \frac{ 899840 }{ 1 } , ~ \pm \frac{ 1349760 }{ 1 } , ~ \pm \frac{ 2699520 }{ 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(8) = 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 - 8} $

$$ \frac{ x^{4}-236x^{3}+17780x^{2}-465088x+2699520 }{ \color{blue}{ x - 8 } } = x^{3}-228x^{2}+15956x-337440 $$

Polynomial $ x^{3}-228x^{2}+15956x-337440 $ can be used to find the remaining roots.

Use the same procedure to find roots of $ x^{3}-228x^{2}+15956x-337440 $

When you get second degree polynomial use step-by-step quadratic equation solver to find two remaining roots.

Polynomial Equations Solver