1. **State the problem:** We are given four polynomials: $$P(x) = 3x^4 + 5x^3 - x^2 + 2$$ $$Q(x) = 5x^3 - 2x^2 + 3x + 4$$ $$R(x) = 1 + x + 2x^2 - 5x^3$$ $$Z(x) = 7 - 2x + x^2$$ We need to simplify the expression: $$Q(x)(R(x) - Z(x)) - P(x)(Q(x) - Z(x))$$ 2. **Write down the formula and rules:** We will use polynomial subtraction and multiplication, and then combine like terms. 3. **Calculate each difference:** $$R(x) - Z(x) = (1 + x + 2x^2 - 5x^3) - (7 - 2x + x^2)$$ $$= 1 + x + 2x^2 - 5x^3 - 7 + 2x - x^2$$ $$= (1 - 7) + (x + 2x) + (2x^2 - x^2) - 5x^3$$ $$= -6 + 3x + x^2 - 5x^3$$ $$Q(x) - Z(x) = (5x^3 - 2x^2 + 3x + 4) - (7 - 2x + x^2)$$ $$= 5x^3 - 2x^2 + 3x + 4 - 7 + 2x - x^2$$ $$= 5x^3 + (-2x^2 - x^2) + (3x + 2x) + (4 - 7)$$ $$= 5x^3 - 3x^2 + 5x - 3$$ 4. **Calculate the products:** $$Q(x)(R(x) - Z(x)) = (5x^3 - 2x^2 + 3x + 4)(-6 + 3x + x^2 - 5x^3)$$ Multiply each term: $$= 5x^3(-6) + 5x^3(3x) + 5x^3(x^2) + 5x^3(-5x^3)$$ $$- 2x^2(-6) - 2x^2(3x) - 2x^2(x^2) - 2x^2(-5x^3)$$ $$+ 3x(-6) + 3x(3x) + 3x(x^2) + 3x(-5x^3)$$ $$+ 4(-6) + 4(3x) + 4(x^2) + 4(-5x^3)$$ Calculate each: $$= -30x^3 + 15x^4 + 5x^5 - 25x^6 + 12x^2 - 6x^3 - 2x^4 + 10x^5 - 18x + 9x^2 + 3x^3 - 15x^4 - 24 + 12x + 4x^2 - 20x^3$$ Combine like terms: $$- 25x^6 + (5x^5 + 10x^5) + (15x^4 - 2x^4 - 15x^4) + (-30x^3 - 6x^3 + 3x^3 - 20x^3) + (12x^2 + 9x^2 + 4x^2) + (-18x + 12x) - 24$$ $$= -25x^6 + 15x^5 + (-2x^4) + (-53x^3) + 25x^2 - 6x - 24$$ $$P(x)(Q(x) - Z(x)) = (3x^4 + 5x^3 - x^2 + 2)(5x^3 - 3x^2 + 5x - 3)$$ Multiply each term: $$= 3x^4(5x^3) + 3x^4(-3x^2) + 3x^4(5x) + 3x^4(-3)$$ $$+ 5x^3(5x^3) + 5x^3(-3x^2) + 5x^3(5x) + 5x^3(-3)$$ $$- x^2(5x^3) - x^2(-3x^2) - x^2(5x) - x^2(-3)$$ $$+ 2(5x^3) + 2(-3x^2) + 2(5x) + 2(-3)$$ Calculate each: $$= 15x^7 - 9x^6 + 15x^5 - 9x^4 + 25x^6 - 15x^5 + 25x^4 - 15x^3 - 5x^5 + 3x^4 - 5x^3 + 3x^2 + 10x^3 - 6x^2 + 10x - 6$$ Combine like terms: $$15x^7 + (-9x^6 + 25x^6) + (15x^5 - 15x^5 - 5x^5) + (-9x^4 + 25x^4 + 3x^4) + (-15x^3 - 5x^3 + 10x^3) + (3x^2 - 6x^2) + 10x - 6$$ $$= 15x^7 + 16x^6 - 5x^5 + 19x^4 - 10x^3 - 3x^2 + 10x - 6$$ 5. **Subtract the two products:** $$Q(x)(R(x) - Z(x)) - P(x)(Q(x) - Z(x))$$ $$= (-25x^6 + 15x^5 - 2x^4 - 53x^3 + 25x^2 - 6x - 24) - (15x^7 + 16x^6 - 5x^5 + 19x^4 - 10x^3 - 3x^2 + 10x - 6)$$ Distribute the minus: $$= -25x^6 + 15x^5 - 2x^4 - 53x^3 + 25x^2 - 6x - 24 - 15x^7 - 16x^6 + 5x^5 - 19x^4 + 10x^3 + 3x^2 - 10x + 6$$ Combine like terms: $$= -15x^7 + (-25x^6 - 16x^6) + (15x^5 + 5x^5) + (-2x^4 - 19x^4) + (-53x^3 + 10x^3) + (25x^2 + 3x^2) + (-6x - 10x) + (-24 + 6)$$ $$= -15x^7 - 41x^6 + 20x^5 - 21x^4 - 43x^3 + 28x^2 - 16x - 18$$ **Final simplified expression:** $$\boxed{-15x^7 - 41x^6 + 20x^5 - 21x^4 - 43x^3 + 28x^2 - 16x - 18}$$