Answer
The inequality is valid for values between -4 and -3 (not including them) i.e. $(-4,-3)$
Work Step by Step
First, we are going to set the right side to zero and factor to find the x-intercepts:
$x^2+7x+12=0$
$(x+4)(x+3)=0$
$x_1=-4$
$x_2=-3$
These are the critical points. We are going to take three values: one less than -4, one between -4 and -3, and one more than -3 to test in the original equation and check if the inequality is true or not:
First test with a value less than -4:
$(-5)^2+7(-5)