Answer
$\emptyset$.
Work Step by Step
Step 1. Given the inequality as $x^2+5x+7\lt0$, form a perfect square for the quadratic: $x^2+5x+7=x^2+5x+(\frac{5}{2})^2-(\frac{5}{2})^2+7=(x+\frac{5}{2})^2+7-\frac{25}{4}=(x+\frac{5}{2})^2+\frac{3}{4}\gt0$
Step 3. As the left side is always positive, the inequality can not be true and the solution set is $\emptyset$.
