<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] if we divide both polynomials we get
$$xa+2\,a+8+1/4\,{\frac {27\,a+3\,b+78}{x-3}}+1/4\,{\frac {a+b-14}{x+1}}$$
and the remainder must be zero. And it must be
$$27a+3b+78=0$$ and $$a+b-14=0$$
</s>