1. **State the problem:** Simplify the expression $$\frac{w^2 - 10w + 21}{18 - 2w^2}$$. 2. **Factor numerator and denominator:** - Numerator: $$w^2 - 10w + 21$$ factors as $$(w - 3)(w - 7)$$ because $-3 \times -7 = 21$ and $-3 + -7 = -10$. - Denominator: $$18 - 2w^2$$ can be factored by taking out the common factor 2: $$2(9 - w^2)$$. 3. **Recognize difference of squares in denominator:** $$9 - w^2 = (3 - w)(3 + w)$$. 4. **Rewrite the expression with factors:** $$\frac{(w - 3)(w - 7)}{2(3 - w)(3 + w)}$$. 5. **Simplify the factor $(3 - w)$:** Note that $$3 - w = -(w - 3)$$. 6. **Substitute and simplify:** $$\frac{(w - 3)(w - 7)}{2 \times -(w - 3)(3 + w)} = \frac{(w - 3)(w - 7)}{-2 (w - 3)(3 + w)}$$. 7. **Cancel common factor $(w - 3)$:** $$\frac{\cancel{(w - 3)}(w - 7)}{-2 \cancel{(w - 3)}(3 + w)} = \frac{w - 7}{-2(3 + w)}$$. 8. **Rewrite denominator:** Since addition is commutative, $$3 + w = w + 3$$, so $$\frac{w - 7}{-2(w + 3)} = -\frac{w - 7}{2(w + 3)}$$. **Final simplified form:** $$-\frac{w - 7}{2(w + 3)}$$