1. **State the problem:** We are given five expressions: $$A = x(x+2) + 2x(x+1)$$ $$B = 8(x+5) - 2(7x+20)$$ $$C = (x-4)x(2x-3) + x - 1$$ $$D = (x+4)x(x-5) - (x+3)$$ $$E = (5x-3)x(x+2) + x(x+3)$$ We need to simplify each expression. 2. **Recall the distributive property:** $$a(b+c) = ab + ac$$ We will apply this to expand and then combine like terms. --- 3. **Simplify A:** $$A = x(x+2) + 2x(x+1) = x^2 + 2x + 2x^2 + 2x = (x^2 + 2x^2) + (2x + 2x) = 3x^2 + 4x$$ --- 4. **Simplify B:** $$B = 8(x+5) - 2(7x+20) = 8x + 40 - 14x - 40 = (8x - 14x) + (40 - 40) = -6x + 0 = -6x$$ --- 5. **Simplify C:** First expand $(x-4)x(2x-3)$: $$ (x-4)x(2x-3) = (x-4)(2x^2 - 3x) = x(2x^2 - 3x) - 4(2x^2 - 3x) = 2x^3 - 3x^2 - 8x^2 + 12x = 2x^3 - 11x^2 + 12x$$ Now add $x - 1$: $$C = 2x^3 - 11x^2 + 12x + x - 1 = 2x^3 - 11x^2 + 13x - 1$$ --- 6. **Simplify D:** First expand $(x+4)x(x-5)$: $$ (x+4)x(x-5) = (x+4)(x^2 - 5x) = x(x^2 - 5x) + 4(x^2 - 5x) = x^3 - 5x^2 + 4x^2 - 20x = x^3 - x^2 - 20x$$ Now subtract $(x+3)$: $$D = x^3 - x^2 - 20x - x - 3 = x^3 - x^2 - 21x - 3$$ --- 7. **Simplify E:** First expand $(5x-3)x(x+2)$: $$ (5x-3)x(x+2) = (5x-3)(x^2 + 2x) = 5x(x^2 + 2x) - 3(x^2 + 2x) = 5x^3 + 10x^2 - 3x^2 - 6x = 5x^3 + 7x^2 - 6x$$ Now add $x(x+3) = x^2 + 3x$: $$E = 5x^3 + 7x^2 - 6x + x^2 + 3x = 5x^3 + 8x^2 - 3x$$ --- **Final answers:** $$A = 3x^2 + 4x$$ $$B = -6x$$ $$C = 2x^3 - 11x^2 + 13x - 1$$ $$D = x^3 - x^2 - 21x - 3$$ $$E = 5x^3 + 8x^2 - 3x$$