1. **State the problem:** Simplify the expression $ \frac{12p^2 - 93}{8p^2 - 29p + 40} $. 2. **Factor numerator and denominator:** - Numerator: $12p^2 - 93 = 3(4p^2 - 31)$ (cannot factor further with integers). - Denominator: Factor $8p^2 - 29p + 40$. 3. **Factor denominator using AC method:** - Multiply $8 \times 40 = 320$. - Find two numbers that multiply to 320 and add to -29: -20 and -9. - Rewrite: $8p^2 - 20p - 9p + 40$. - Group: $(8p^2 - 20p) + (-9p + 40)$. - Factor each group: $4p(2p - 5) - 8(2p - 5)$. - Factor out common binomial: $(4p - 8)(2p - 5)$. - Simplify $4p - 8 = 4(p - 2)$. - So denominator factors as $4(p - 2)(2p - 5)$. 4. **Rewrite the fraction:** $$\frac{3(4p^2 - 31)}{4(p - 2)(2p - 5)}$$ 5. **Check for common factors:** - Numerator $4p^2 - 31$ does not factor further and shares no common factors with denominator. 6. **Final simplified form:** $$\frac{3(4p^2 - 31)}{4(p - 2)(2p - 5)}$$ This is the simplest form since numerator and denominator share no common factors.