1. **State the problem:** Simplify the expression $$\frac{16x^2 + 24x + 9}{3x^2 + 13x - 10} \cdot \frac{6x^2 + 25x - 25}{4x^2 - 13x - 12}$$ and verify the factorization $$(4x + 3)(4x + 3) \cdot (x + 5)(x + 5)$$ 2. **Factor each polynomial:** - Numerator 1: $16x^2 + 24x + 9 = (4x + 3)^2$ - Denominator 1: $3x^2 + 13x - 10$ Find factors of $3 \times (-10) = -30$ that sum to 13: 15 and -2 So, $$3x^2 + 15x - 2x - 10 = 3x(x + 5) - 2(x + 5) = (3x - 2)(x + 5)$$ - Numerator 2: $6x^2 + 25x - 25$ Factors of $6 \times (-25) = -150$ that sum to 25: 30 and -5 So, $$6x^2 + 30x - 5x - 25 = 6x(x + 5) - 5(x + 5) = (6x - 5)(x + 5)$$ - Denominator 2: $4x^2 - 13x - 12$ Factors of $4 \times (-12) = -48$ that sum to -13: -16 and 3 So, $$4x^2 - 16x + 3x - 12 = 4x(x - 4) + 3(x - 4) = (4x + 3)(x - 4)$$ 3. **Rewrite the expression with factors:** $$\frac{(4x + 3)^2}{(3x - 2)(x + 5)} \cdot \frac{(6x - 5)(x + 5)}{(4x + 3)(x - 4)}$$ 4. **Simplify by canceling common factors:** - Cancel one $(4x + 3)$ from numerator and denominator - Cancel $(x + 5)$ from numerator and denominator Resulting expression: $$\frac{4x + 3}{3x - 2} \cdot \frac{6x - 5}{x - 4} = \frac{(4x + 3)(6x - 5)}{(3x - 2)(x - 4)}$$ 5. **Multiply the numerators and denominators:** Numerator: $$ (4x + 3)(6x - 5) = 24x^2 - 20x + 18x - 15 = 24x^2 - 2x - 15$$ Denominator: $$ (3x - 2)(x - 4) = 3x^2 - 12x - 2x + 8 = 3x^2 - 14x + 8$$ 6. **Final simplified expression:** $$\frac{24x^2 - 2x - 15}{3x^2 - 14x + 8}$$ --- **Verification of given factorization:** $$(4x + 3)(4x + 3) \cdot (x + 5)(x + 5) = (4x + 3)^2 (x + 5)^2$$ This matches the factorization of the numerators combined. --- **Answer:** $$\boxed{\frac{24x^2 - 2x - 15}{3x^2 - 14x + 8}}$$