1. **State the problem:** Simplify the expression $$\frac{32x}{x^2 + x - 72} \div \frac{16x - 8}{2x^2 - 17x + 8}$$ 2. **Rewrite division as multiplication by reciprocal:** $$\frac{32x}{x^2 + x - 72} \times \frac{2x^2 - 17x + 8}{16x - 8}$$ 3. **Factor all polynomials:** - Factor denominator $x^2 + x - 72$: $$x^2 + x - 72 = (x + 9)(x - 8)$$ - Factor numerator $16x - 8$: $$16x - 8 = 8(2x - 1)$$ - Factor numerator $2x^2 - 17x + 8$: Find factors of $2 \times 8 = 16$ that sum to $-17$: $-16$ and $-1$ $$2x^2 - 17x + 8 = 2x^2 - 16x - x + 8 = 2x(x - 8) - 1(x - 8) = (2x - 1)(x - 8)$$ 4. **Substitute factored forms:** $$\frac{32x}{(x + 9)(x - 8)} \times \frac{(2x - 1)(x - 8)}{8(2x - 1)}$$ 5. **Cancel common factors:** - Cancel $(x - 8)$: $$\frac{32x}{(x + 9)\cancel{(x - 8)}} \times \frac{(2x - 1)\cancel{(x - 8)}}{8(2x - 1)}$$ - Cancel $(2x - 1)$: $$\frac{32x}{(x + 9)} \times \frac{\cancel{(2x - 1)}}{8\cancel{(2x - 1)}}$$ 6. **Multiply remaining terms:** $$\frac{32x}{x + 9} \times \frac{1}{8} = \frac{32x}{8(x + 9)}$$ 7. **Simplify the fraction:** $$\frac{\cancel{32}^4 x}{\cancel{8}^1 (x + 9)} = \frac{4x}{x + 9}$$ **Final answer:** $$\boxed{\frac{4x}{x + 9}}$$