Answer
$\emptyset$.
Work Step by Step
Step 1. Given the inequality as $4x^2+3x+1\lt0$, form a perfect square for the quadratic: $4x^2+3x+1=4(x^2+\frac{3}{4}x)+1=4(x^2+\frac{3}{4}x+(\frac{3}{8})^2-(\frac{3}{8})^2)+1=4(x^2+\frac{3}{8})^2-(4)(\frac{3}{8})^2+1=4(x^2+\frac{3}{8})^2+\frac{7}{16}\gt0$
Step 2. As the left side is always positive, the inequality can not be true and the solution set is $\emptyset$.
