Find the zeros of the following quadratic polynomials $x^2-2x-8$ and verify the relationship between the zeros and the coefficients.

Step 1: Form the equation

The given polynomial is $x^2-2x-8$.

We know that the zeroes of a polynomial are evaluated by equating them with zero.

$\therefore p\left(x\right)=0$

$\Rightarrow x^2-2x-8=0$

Step 2: Solve to find the zeroes

$x^2-2x-8=0$

$\Rightarrow$ $x^2-4x+2x-8=0$

$\Rightarrow$$x\left(x-4\right)+2\left(x-4\right)=0$

$\Rightarrow$ $\left(x-4\right)\left(x+2\right)=0$

$\Rightarrow$ $x=4,-2$

Step 3: Verification

We know that for a given polynomial $a{x}^2+bx+c$,

Sum of the zeroes$=-\frac{b}{a}$ and product of the roots$=\frac{c}{a}$

Sum of the zeroes $=4-2=2$

Again, $-\frac{b}{a}=-\frac{\left(-2\right)}{1}=2$

Product of the zeroes $=4\left(-2\right)=-8$

Again, $\frac{c}{a}=\frac{8}{1}=8$

Thus, the relationship between the zeroes and the coefficients of the polynomial is verified.

Hence, the zeroes of this polynomial are $4$ and $-2$.