Let p(x) = x^{3} - 3x^{2} -12x +19 and q(x) = x^{2} + 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, x^{3} - 3x^{2} -12x +19 is divisible by x^{2} + x -6 when 2x+5 is added to it.