1. **Problem statement:** Expand and simplify each expression. 2. **Part a:** Expression: $3(6x^2 - 2x - 1) - 4(2x^2 - 3x + 5)$ Use distributive property: $a(b + c) = ab + ac$ Expand: $$3 \times 6x^2 = 18x^2$$ $$3 \times (-2x) = -6x$$ $$3 \times (-1) = -3$$ $$-4 \times 2x^2 = -8x^2$$ $$-4 \times (-3x) = +12x$$ $$-4 \times 5 = -20$$ Combine: $$18x^2 - 6x - 3 - 8x^2 + 12x - 20$$ Group like terms: $$ (18x^2 - 8x^2) + (-6x + 12x) + (-3 - 20)$$ Simplify: $$10x^2 + 6x - 23$$ 3. **Part b:** Expression: $5k(k + 7) - (k^2 + 4)$ Expand: $$5k \times k = 5k^2$$ $$5k \times 7 = 35k$$ Expression becomes: $$5k^2 + 35k - k^2 - 4$$ Group like terms: $$(5k^2 - k^2) + 35k - 4$$ Simplify: $$4k^2 + 35k - 4$$ 4. **Part c:** Expression: $\frac{1}{3}(6w + 9) - \frac{3}{4}(8w - 12)$ Expand: $$\frac{1}{3} \times 6w = 2w$$ $$\frac{1}{3} \times 9 = 3$$ $$\frac{3}{4} \times 8w = 6w$$ $$\frac{3}{4} \times (-12) = -9$$ Expression becomes: $$2w + 3 - 6w + 9$$ Group like terms: $$(2w - 6w) + (3 + 9)$$ Simplify: $$-4w + 12$$ 5. **Part d:** Expression: $-3x(2x + 3y - 5)$ Expand: $$-3x \times 2x = -6x^2$$ $$-3x \times 3y = -9xy$$ $$-3x \times (-5) = +15x$$ Final expression: $$-6x^2 - 9xy + 15x$$ **Final answers:** - a) $10x^2 + 6x - 23$ - b) $4k^2 + 35k - 4$ - c) $-4w + 12$ - d) $-6x^2 - 9xy + 15x$