To simplify, we will have to factor the numerator of the first fraction and the denominator of the second fraction.
\frac { x ^ { 2} + x - 12} { color(blue)(x) color(green)(y ^ { 2}) } \cdot \frac { color(blue)(x ^ { 7})color(green)( y )} { x ^ { 2} + 7x + 12}
=\frac { cancel((x+4))(x-3)} { color(blue)(x) color(green)(y ^ { 2}) } \cdot \frac { color(blue)(x ^ { 7})color(green)( y) } { cancel((x+4))(x+3)}
We see that the (x+4) from the top and the bottom cancel, but also an xy factor (from the denominator of the first fraction and the numerator of the second fraction).
Thus, this simplifies to:
=\frac { (x-3)} { y} \cdot \frac { x ^ { 6}} { (x+3)}
=\frac { x ^ { 6}(x-3)} { y(x+3)}