1. **State the problem:** Simplify the rational expression $$\frac{y^2 - 3y - 28}{y^2 + 12y + 32}$$ and find all values of $y$ that must be excluded from the domain. 2. **Factor numerator and denominator:** - Numerator: $y^2 - 3y - 28$ - Find two numbers that multiply to $-28$ and add to $-3$: these are $-7$ and $4$. - So, numerator factors as $$(y - 7)(y + 4)$$. - Denominator: $y^2 + 12y + 32$ - Find two numbers that multiply to $32$ and add to $12$: these are $8$ and $4$. - So, denominator factors as $$(y + 8)(y + 4)$$. 3. **Write the expression with factors:** $$\frac{(y - 7)(y + 4)}{(y + 8)(y + 4)}$$ 4. **Simplify by canceling common factors:** - The factor $(y + 4)$ appears in numerator and denominator, so it can be canceled: $$\frac{(y - 7)\cancel{(y + 4)}}{(y + 8)\cancel{(y + 4)}}$$ - Simplified expression is: $$\frac{y - 7}{y + 8}$$ 5. **Find excluded values from the domain:** - The original denominator cannot be zero, so solve: $$y^2 + 12y + 32 = 0$$ - Factored form: $$(y + 8)(y + 4) = 0$$ - So, $y = -8$ or $y = -4$ are excluded. 6. **Check for restrictions after simplification:** - Even though $(y + 4)$ was canceled, $y = -4$ is still excluded because it makes the original denominator zero. **Final answer:** - Simplified expression: $$\frac{y - 7}{y + 8}$$ - Excluded values: $$y \neq -8, -4$$