Answer
$$x=-\frac{1}{2},\:x=-\frac{3}{2}$$
Work Step by Step
In order to solve this problem, we use the method of completing the square. Doing this, we find:
$$ x^2+2x=-\frac{3}{4} \\ x^2+2x+1^2=\frac{1}{4}\\ \left(x+1\right)^2=\frac{1}{4}\\ x=-\frac{1}{2},\:x=-\frac{3}{2}$$