1. **State the problem:** Factor the expressions $12 + 30$ and $36 + 63$ by finding their greatest common factors (GCF). 2. **Formula and property used:** Use the distributive property: $$a \times b + a \times c = a(b + c)$$ where $a$ is the GCF. 3. **Find the GCF of $12$ and $30$:** - Factors of $12$: $1, 2, 3, 4, 6, 12$ - Factors of $30$: $1, 2, 3, 5, 6, 10, 15, 30$ - Common factors: $1, 2, 3, 6$ - Greatest common factor: $6$ 4. **Rewrite each term as a product of the GCF:** $$12 = 6 \times 2$$ $$30 = 6 \times 5$$ 5. **Apply the distributive property:** $$12 + 30 = 6 \times 2 + 6 \times 5 = 6(2 + 5)$$ 6. **Simplify inside the parentheses:** $$6(2 + 5) = 6 \times 7 = 42$$ 7. **Find the GCF of $36$ and $63$:** - Factors of $36$: $1, 2, 3, 4, 6, 9, 12, 18, 36$ - Factors of $63$: $1, 3, 7, 9, 21, 63$ - Common factors: $1, 3, 9$ - Greatest common factor: $9$ 8. **Rewrite each term as a product of the GCF:** $$36 = 9 \times 4$$ $$63 = 9 \times 7$$ 9. **Apply the distributive property:** $$36 + 63 = 9 \times 4 + 9 \times 7 = 9(4 + 7)$$ 10. **Simplify inside the parentheses:** $$9(4 + 7) = 9 \times 11 = 99$$ **Final factored expressions:** $$12 + 30 = 6(2 + 5)$$ $$36 + 63 = 9(4 + 7)$$