1. **Problem statement:** Factor the quadratic expressions given in part 21, starting with (a) 4y^2 - 20y - 56. 2. **Formula and rules:** To factor a quadratic expression of the form $ax^2 + bx + c$, we look for two numbers that multiply to $a \times c$ and add to $b$. 3. **Step-by-step for (a) 4y^2 - 20y - 56:** - First, find the greatest common factor (GCF) of all terms: GCF is 4. - Factor out 4: $$4(y^2 - 5y - 14)$$ - Now factor inside the parentheses: find two numbers that multiply to $-14$ and add to $-5$. These are $-7$ and $2$. - So, $$y^2 - 5y - 14 = (y - 7)(y + 2)$$ - Final factorization: $$4(y - 7)(y + 2)$$ 4. **Step-by-step for (b) -3m^2 - 18m - 24:** - GCF is $-3$ (to keep the leading coefficient positive inside parentheses). - Factor out $-3$: $$-3(m^2 + 6m + 8)$$ - Find two numbers that multiply to $8$ and add to $6$: $2$ and $4$. - Factor inside: $$(m + 2)(m + 4)$$ - Final factorization: $$-3(m + 2)(m + 4)$$ 5. **Step-by-step for (c) 4x^2 + 4x - 48:** - GCF is 4. - Factor out 4: $$4(x^2 + x - 12)$$ - Find two numbers that multiply to $-12$ and add to $1$: $4$ and $-3$. - Factor inside: $$(x + 4)(x - 3)$$ - Final factorization: $$4(x + 4)(x - 3)$$ 6. **Step-by-step for (d) 10x^2 + 80x + 120:** - GCF is 10. - Factor out 10: $$10(x^2 + 8x + 12)$$ - Find two numbers that multiply to $12$ and add to $8$: $6$ and $2$. - Factor inside: $$(x + 6)(x + 2)$$ - Final factorization: $$10(x + 6)(x + 2)$$ 7. **Step-by-step for (e) -5n^2 + 40n - 35:** - GCF is $-5$. - Factor out $-5$: $$-5(n^2 - 8n + 7)$$ - Find two numbers that multiply to $7$ and add to $-8$: $-7$ and $-1$. - Factor inside: $$(n - 7)(n - 1)$$ - Final factorization: $$-5(n - 7)(n - 1)$$ 8. **Step-by-step for (f) 7c^2 - 35c + 42:** - GCF is 7. - Factor out 7: $$7(c^2 - 5c + 6)$$ - Find two numbers that multiply to $6$ and add to $-5$: $-3$ and $-2$. - Factor inside: $$(c - 3)(c - 2)$$ - Final factorization: $$7(c - 3)(c - 2)$$ **Summary:** - (a) $4(y - 7)(y + 2)$ - (b) $-3(m + 2)(m + 4)$ - (c) $4(x + 4)(x - 3)$ - (d) $10(x + 6)(x + 2)$ - (e) $-5(n - 7)(n - 1)$ - (f) $7(c - 3)(c - 2)$ This method helps break down quadratics into simpler binomials, making them easier to work with.