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What must be added to x3 - 3x2 -12x +19 so that the result is exactly divisibly by x2 + x -6 ?

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Let p(x) = x3 - 3x2 -12x +19 and q(x) = x2 + x -6.

By division algorithm, when p(x) is divided by q(x), the remainder is a linear expression in x

So, let r(x) = ax +b is added to p(x) so that p(x) + r(x) is divisible by q(x).

Hence, x3 - 3x2 -12x +19 is divisible by x2 + x -6 when 2x+5 is added to it.

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