Factorize the quadratic.
[tex]2x^2 + 5x + 3 = (2x + 3) (x + 1)[/tex]
We have [tex]|ab| = |a||b|[/tex], so
[tex]|2x^2 + 5x + 3| = |2x + 3| |x + 1| > 0[/tex]
Now, both [tex]|2x+3|\ge0[/tex] and [tex]|x+1|\ge0[/tex] (since the absolute value of any number cannot be negative), so we just need to worry about when the left side is exactly zero. This happens for
[tex](2x + 3) (x + 1) = 0 \implies 2x+3 = 0\text{ or }x+1 = 0 \\\\ \implies x = -\dfrac32 \text{ or } x = -1[/tex]
So the solution to the inequality is the set
[tex]\left\{x \in \Bbb R \mid x\neq-\dfrac32 \text{ and } x\neq-1\right\}[/tex]
