Answer
$\frac{1}{(x-4)}$
Work Step by Step
We will simplify the expression by first factoring the quadratic equations in both the numerator and the denominators and then cancelling out the resultant common factors:
$\frac{x-2}{x^{2}+x-6}\times\frac{x^{2}+6x+9}{x^{2}-x-12}$
=$\frac{x-2}{x^{2}-2x+3x-6}\times\frac{x^{2}+3x+3x+9}{x^{2}+3x-4x-12}$
=$\frac{x-2}{x(x-2)+3(x-2)}\times\frac{x(x+3)+3(x+3)}{x(x+3)-4(x+3)}$
=$\frac{x-2}{(x-2)(x+3)}\times\frac{(x+3)(x+3)}{(x+3)(x-4)}$
=$\frac{x-2}{(x-2)(x+3)}\times\frac{(x+3)}{(x-4)}$
=$\frac{x-2}{(x-2)}\times\frac{1}{(x-4)}$
=$\frac{1}{1}\times\frac{1}{(x-4)}$
=$\frac{1}{(x-4)}$
