$ax^3+8x^2+bx+6$ 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 $x^2-2x-3$.

Let $f(x)=ax^3+8x^2+bx+6$, and let $g(x)=x^2-2x-3$. Since the question says that $g(x)|f(x)$, then write $f(x)=g(x)(px+q)$. Note the linear polynomial, we did that because dividing $f(x)$ by $g(x)$ will result in degree $1$. Now, expand the LHS and RHS, and you get something like this:
$$ax^3+8x^2+bx+6=(x^2-2x-3)(px+q)=px^3+x^2(q-2p)-x(2q+3p)-3q$$
Clearly, $q=-2,$ and $p=-5$. Now get the values of $a$ and $b$ by comparing the co-efficients. 
