Divide and, if possible, simplify. \( \frac{w^{2}-6 w+9}{3 w^{2}-2 w-33} \div \frac{2 w^{2}-5 w-3}{w^{2}-5 w-24} \)

To divide fractions, we multiply by the reciprocal of the second fraction. Thus, we rewrite the expression as: \[ \frac{w^{2}-6 w+9}{3 w^{2}-2 w-33} \times \frac{w^{2}-5 w-24}{2 w^{2}-5 w-3} \] Now, let's simplify each part. 1. **Factor \(w^{2}-6w+9\):** \( w^{2}-6w+9 = (w-3)(w-3) = (w-3)^{2} \) 2. **Factor \(3w^{2}-2w-33\):** We look for two numbers that multiply to \(3 \cdot (-33) = -99\) and add to \(-2\). The numbers are \(9\) and \(-11\). \[ 3w^{2}+9w-11w-33 = 3w(w+3)-11(w+3) = (3w-11)(w+3) \] 3. **Factor \(2w^{2}-5w-3\):** We look for two numbers that multiply to \(2 \cdot (-3) = -6\) and add to \(-5\). The numbers are \(-6\) and \(1\). \[ 2w^{2}-6w+w-3 = 2w(w-3)+1(w-3) = (2w+1)(w-3) \] 4. **Factor \(w^{2}-5w-24\):** We look for two numbers that multiply to \(-24\) and add to \(-5\). The numbers are \(-8\) and \(3\). \[ w^{2}-8w+3w-24 = w(w-8)+3(w-8) = (w-8)(w+3) \] Now substituting these factorizations back into the expression, we get: \[ \frac{(w-3)^{2}}{(3w-11)(w+3)} \times \frac{(w-8)(w+3)}{(2w+1)(w-3)} \] Next, let’s multiply the fractions: \[ \frac{(w-3)^{2} (w-8)(w+3)}{(3w-11)(w+3)(2w+1)(w-3)} \] Now cancel out the common factors \( (w-3) \) and \( (w+3) \): \[ \frac{(w-3)(w-8)}{(3w-11)(2w+1)} \] Thus, the simplified expression is: \[ \frac{(w-3)(w-8)}{(3w-11)(2w+1)} \] This is the final simplified result!