Question: Factor the trinomial. (5x + 3)^2 + 8(5x + 3) + 15
25x^2 + 40x + 48
1. **State the problem:** Factor the trinomial $$ (5x + 3)^2 + 8(5x + 3) + 15 $$ and verify if it equals $$ 25x^2 + 40x + 48 $$.
2. **Rewrite the expression:** Let $$ u = 5x + 3 $$, then the expression becomes $$ u^2 + 8u + 15 $$.
3. **Factor the quadratic in terms of $$ u $$:** We look for two numbers that multiply to $$ 15 $$ and add to $$ 8 $$. These numbers are $$ 3 $$ and $$ 5 $$.
4. **Write the factored form:** $$ u^2 + 8u + 15 = (u + 3)(u + 5) $$
5. **Substitute back $$ u = 5x + 3 $$:**
$$ (5x + 3 + 3)(5x + 3 + 5) = (5x + 6)(5x + 8) $$
6. **Verify by expanding:**
$$ (5x + 6)(5x + 8) = 25x^2 + 40x + 48 $$ which matches the given trinomial.
**Final answer:**
$$ \boxed{(5x + 6)(5x + 8)} $$