1. **Problem statement:** Simplify each algebraic expression by applying the Distributive Property and combining like terms. 2. **Formula and rules:** - Distributive Property: $$a(b + c) = ab + ac$$ - Combine like terms by adding coefficients of the same variable. 3. **Part a:** $$2(c + 1) + 3(c + 2)$$ - Apply distributive property: $$2c + 2 + 3c + 6$$ - Combine like terms: $$2c + 3c + 2 + 6 = 5c + 8$$ 4. **Part b:** $$4a + 2(3a + 8)$$ - Apply distributive property: $$4a + 6a + 16$$ - Combine like terms: $$4a + 6a + 16 = 10a + 16$$ 5. **Part c:** $$5(6x + 2) + 3(2x + 1)$$ - Apply distributive property: $$30x + 10 + 6x + 3$$ - Combine like terms: $$30x + 6x + 10 + 3 = 36x + 13$$ 6. **Part d:** $$4(5x + 1) + 3(x + 2)$$ - Apply distributive property: $$20x + 4 + 3x + 6$$ - Combine like terms: $$20x + 3x + 4 + 6 = 23x + 10$$ 7. **Optional challenge e:** $$3(5x + 4y + 9z + 3) + 4(3x + 5y + z + 5) + 16 \left(\frac{3}{4}x + \frac{5}{8}y + \frac{7}{16}z + \frac{1}{2}\right)$$ - Apply distributive property: $$15x + 12y + 27z + 9 + 12x + 20y + 4z + 20 + 16 \times \frac{3}{4}x + 16 \times \frac{5}{8}y + 16 \times \frac{7}{16}z + 16 \times \frac{1}{2}$$ - Simplify multiplication: $$15x + 12y + 27z + 9 + 12x + 20y + 4z + 20 + 12x + 10y + 7z + 8$$ - Combine like terms: $$ (15x + 12x + 12x) + (12y + 20y + 10y) + (27z + 4z + 7z) + (9 + 20 + 8)$$ $$= 39x + 42y + 38z + 37$$