1. **Problem:** Expand and simplify the expression $a (2x + 1)^2$. 2. **Formula and rules:** To expand a binomial squared, use the formula $ (A + B)^2 = A^2 + 2AB + B^2 $. 3. **Step-by-step expansion:** $$ (2x + 1)^2 = (2x)^2 + 2 \times 2x \times 1 + 1^2 $$ $$ = 4x^2 + 4x + 1 $$ 4. **Multiply by $a$:** $$ a (2x + 1)^2 = a (4x^2 + 4x + 1) $$ $$ = 4a x^2 + 4a x + a $$ 5. **Final answer:** $$ \boxed{4a x^2 + 4a x + a} $$ This completes the expansion and simplification of $a (2x + 1)^2$.