Answer
All real numbers solve the inequality.
Work Step by Step
$x^2+3x+8\gt0\\(x+1.5)^2-2.25+8\gt0\\(x+1.5)^2+6.75\gt0\\(x+1.5)^2\gt-6.75$
but we know that $a^2\geq0$ for all $a$ so all numbers satisfy this inequality. Thus the solution set is the whole set of real numbers.
