1. **State the problem:** Simplify the expression $ (9x + 5)^2 $. 2. **Formula used:** The square of a binomial $ (a + b)^2 $ is given by the formula: $$ (a + b)^2 = a^2 + 2ab + b^2 $$ where $a = 9x$ and $b = 5$. 3. **Apply the formula:** $$ (9x + 5)^2 = (9x)^2 + 2 \times 9x \times 5 + 5^2 $$ 4. **Calculate each term:** $$ (9x)^2 = 81x^2 $$ $$ 2 \times 9x \times 5 = 90x $$ $$ 5^2 = 25 $$ 5. **Combine all terms:** $$ 81x^2 + 90x + 25 $$ 6. **Final answer:** The simplified form of $ (9x + 5)^2 $ is $$ \boxed{81x^2 + 90x + 25} $$