1. Problem: Multiply each pair of binomials/trinomials. (a) Multiply $(3x - 2y)(4x + 3y)$ Use distributive property (FOIL method): $$ (3x)(4x) + (3x)(3y) - (2y)(4x) - (2y)(3y) = 12x^2 + 9xy - 8xy - 6y^2 $$ Simplify like terms: $$ 12x^2 + (9xy - 8xy) - 6y^2 = 12x^2 + xy - 6y^2 $$ 2. (b) Multiply $(2y - 1)(3 + 2y)$ Distribute each term: $$ (2y)(3) + (2y)(2y) - 1(3) - 1(2y) = 6y + 4y^2 - 3 - 2y $$ Simplify like terms: $$ 4y^2 + (6y - 2y) - 3 = 4y^2 + 4y - 3 $$ 3. (c) Multiply $(7x + 2y)(7x - 2y)$ Use difference of squares formula: $$ (7x)^2 - (2y)^2 = 49x^2 - 4y^2 $$ 4. (d) Multiply $(x - x + z)(x + y)$ Simplify inside first parentheses: $x - x + z = z$ So, $$ z(x + y) = zx + zy $$ 5. (e) Multiply $(3x + 2)(x^2 - 2x + 1)$ Distribute each term: $$ 3x(x^2 - 2x + 1) + 2(x^2 - 2x + 1) = 3x^3 - 6x^2 + 3x + 2x^2 - 4x + 2 $$ Simplify like terms: $$ 3x^3 + (-6x^2 + 2x^2) + (3x - 4x) + 2 = 3x^3 - 4x^2 - x + 2 $$ 6. (f) Multiply $(x^2 - 2x)(3x^2 + 2x + 3)$ Distribute the terms: $$ x^2(3x^2 + 2x + 3) - 2x(3x^2 + 2x + 3) = 3x^4 + 2x^3 + 3x^2 - 6x^3 - 4x^2 - 6x $$ Simplify like terms: $$ 3x^4 + (2x^3 - 6x^3) + (3x^2 - 4x^2) - 6x = 3x^4 - 4x^3 - x^2 - 6x $$