Question:
Find the value of $a$ for which $(x+1)$ is a factor of $\left(a x^{3}+x^{2}-2 x+4 a-9\right)$.
Solution:
Let $f(x)=a x^{3}+x^{2}-2 x+4 a-9$
It is given that (x + 1) is a factor of f(x).
Using factor theorem, we have
$f(-1)=0$
$\Rightarrow a \times(-1)^{3}+(-1)^{2}-2 \times(-1)+4 a-9=0$
$\Rightarrow-a+1+2+4 a-9=0$
$\Rightarrow 3 a-6=0$
$\Rightarrow 3 a=6$
$\Rightarrow a=2$
Thus, the value of a is 2.