Question

98 POINTS ASAP

Answer 1

The rational root test leads to two roots of $f(x)$, $f(-1)=f(-2)=0$, which means both $x+1$ and $x+2$ are factors.

$\dfrac{x^4 -3x^3 - 5x^2 + 21x + 22}{(x+1)(x+2)}=x^2-6x+11$

The discriminant of the remaining quadratic is negative $((-6)^2-4(11)=-8)$ and so it has no real roots. So

$f(x)=(x+1)(x+2)(x^2-6x+11)$