<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] by using the long division we get
$$\frac{ax^3+8x^2+bx+6}{x^2-2x-3}=ax+(8+2a)+\frac{(7a+b+16)x+6a+30}{x^2-2x-3}$$
now the reminder should be zero
$$7a+b+16=0\tag 1$$
$$6a+30=0\tag2$$
$$a=-5, b=19$$
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