If polynomial is divisible by quadratic, find values of $a$ and $b$ Equation is $z^4+(a+b)z^3+4az^2+(a+b+32)z+45$ which is divisible by $z^2+6z+9$.
We're meant to find values of $a$ and $b$ and then solve the equation itself!
I figured $z^2+6z+9=(z+3)^2$, but was wondering if this automatically implies that it is a double root(?) If so, would differentiating the equation and do the normal simultaneous equations thing suitable for this question?

use that $${z}^{2}+ \left( a+b-6 \right) z-2\,a-6\,b+27+{\frac {4\,a+28\,b-76}{z+
3}}+{\frac {6\,a-30\,b+30}{ \left( z+3 \right) ^{2}}}=\frac{z^4+(a+b)z^3+4az^2+(a+b+32)z+45}{z^2+6z+9}
$$
