Find the zeros of the following quadratic polynomials and verify the relationship between the zeros and the coefficients:

Let $f(x)=5 x^{2}+10 x$

To find the zeroes, we put $f(x)=0$

$\Rightarrow 5 x^{2}+10 x=0$

$\Rightarrow 5 x(x+2)=0$

$\Rightarrow(x)(x+2)=0$

$\Rightarrow(x)=0$ or $(x+2)=0$

$\Rightarrow x=0,-2$

Hence, all the zeroes of the polynomial $f(x)$ are 0 and $-2$.

Now,

$f(0)=5(0)^{2}+10(0)$

$=0+0$

$=0$

$f(-2)=5(-2)^{2}+10(-2)$

$=5(4)-20$

$=20-20$

$=0$

Hence, the relationship between the zeros and the coefficients is verified.