I3.2 Given that $x^{2}+a x+b$ is a common factor of $2 x^{3}+5 x^{2}+24 x+11$ and $x^{3}+P x-22$. If $Q=a+b$, find the value of $Q$.
Let $f(x)=2 x^{3}+5 x^{2}+24 x+11 ; g(x)=x^{3}+7 x-22$
$g(2)=8+14-22=0 \Rightarrow x-2$ is a factor
By division $g(x)=(x-2)\left(x^{2}+2 x+11\right) ; f(x)=(2 x+1)\left(x^{2}+2 x+11\right)$
$$
a=2, b=11 ; Q=a+b=13
$$

Method 2
Let $f(x)=2 x^{3}+5 x^{2}+24 x+11=\left(x^{2}+a x+b\right)(c x+d)$
$$
\begin{array}{l}
g(x)=x^{3}+7 x-22=\left(x^{2}+a x+b\right)(p x+q) \\
f(x)-2 g(x)=2 x^{3}+5 x^{2}+24 x+11-2\left(x^{3}+7 x-22\right) \equiv\left(x^{2}+a x+b\right)[(c-2 d) x+d-2 q] \\
5 x^{2}+10 x+55 \equiv\left(x^{2}+a x+b\right)[(c-2 d) x+d-2 q]
\end{array}
$$

By comparing coefficients of $x^{3}$ and $x^{2}$ on both sides:
$$
\begin{array}{l}
c=2 d \text { and } d-2 q=5 \\
5 x^{2}+10 x+55 \equiv 5\left(x^{2}+a x+b\right) \\
a=2, b=11 \\
Q=a+b=13
\end{array}
$$