<s>[INST] $ax^3+8x^2+bx+6$ is exactly divisible by $x^2-2x-3$, find the values of $a$ and $b$ Find the values of $a$ and $b$ for which the polynomial $ax^3+8x^2+bx+6$ is divisible by $x^2-2x-3$.
 [/INST] $$ax^3+8x^2+bx+6=(ax+c) (x^2-2x-3)=ax^3+(c-2a)x^2+(-2c-3a)x-3c$$
$$\begin{align}6&=-3c\Rightarrow c=-2\\8&=c-2a\Rightarrow 2a=c-8=-10\Rightarrow \color{red}{a=-5}\\b&=-2c-3a=4+15=\color{red}{19}\end{align}$$
Using this approach, you will also find out the third root is given by $ax+c=0\Rightarrow x=-\frac{2}{5}$
Also $x^2-2x-3x=(x-3)(x+1)$ gives $x=3$ and $x=-1$.
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