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Simplifying x(x + -1)(x + 2)(x + 3) + 1 = (x2 + 3x + 1)(x2 + 3x + 1) Reorder the terms: x(-1 + x)(x + 2)(x + 3) + 1 = (x2 + 3x + 1)(x2 + 3x + 1) Reorder the terms: x(-1 + x)(2 + x)(x + 3) + 1 = (x2 + 3x + 1)(x2 + 3x + 1) Reorder the terms: x(-1 + x)(2 + x)(3 + x) + 1 = (x2 + 3x + 1)(x2 + 3x + 1) Multiply (-1 + x) * (2 + x) x(-1(2 + x) + x(2 + x))(3 + x) + 1 = (x2 + 3x + 1)(x2 + 3x + 1) x((2 * -1 + x * -1) + x(2 + x))(3 + x) + 1 = (x2 + 3x + 1)(x2 + 3x + 1) x((-2 + -1x) + x(2 + x))(3 + x) + 1 = (x2 + 3x + 1)(x2 + 3x + 1) x(-2 + -1x + (2 * x + x * x))(3 + x) + 1 = (x2 + 3x + 1)(x2 + 3x + 1) x(-2 + -1x + (2x + x2))(3 + x) + 1 = (x2 + 3x + 1)(x2 + 3x + 1) Combine like terms: -1x + 2x = 1x x(-2 + 1x + x2)(3 + x) + 1 = (x2 + 3x + 1)(x2 + 3x + 1) Multiply (-2 + 1x + x2) * (3 + x) x(-2(3 + x) + 1x * (3 + x) + x2(3 + x)) + 1 = (x2 + 3x + 1)(x2 + 3x + 1) x((3 * -2 + x * -2) + 1x * (3 + x) + x2(3 + x)) + 1 = (x2 + 3x + 1)(x2 + 3x + 1) x((-6 + -2x) + 1x * (3 + x) + x2(3 + x)) + 1 = (x2 + 3x + 1)(x2 + 3x + 1) x(-6 + -2x + (3 * 1x + x * 1x) + x2(3 + x)) + 1 = (x2 + 3x + 1)(x2 + 3x + 1) x(-6 + -2x + (3x + 1x2) + x2(3 + x)) + 1 = (x2 + 3x + 1)(x2 + 3x + 1) x(-6 + -2x + 3x + 1x2 + (3 * x2 + x * x2)) + 1 = (x2 + 3x + 1)(x2 + 3x + 1) x(-6 + -2x + 3x + 1x2 + (3x2 + x3)) + 1 = (x2 + 3x + 1)(x2 + 3x + 1) Combine like terms: -2x + 3x = 1x x(-6 + 1x + 1x2 + 3x2 + x3) + 1 = (x2 + 3x + 1)(x2 + 3x + 1) Combine like terms: 1x2 + 3x2 = 4x2 x(-6 + 1x + 4x2 + x3) + 1 = (x2 + 3x + 1)(x2 + 3x + 1) (-6 * x + 1x * x + 4x2 * x + x3 * x) + 1 = (x2 + 3x + 1)(x2 + 3x + 1) (-6x + 1x2 + 4x3 + x4) + 1 = (x2 + 3x + 1)(x2 + 3x + 1) Reorder the terms: 1 + -6x + 1x2 + 4x3 + x4 = (x2 + 3x + 1)(x2 + 3x + 1) Reorder the terms: 1 + -6x + 1x2 + 4x3 + x4 = (1 + 3x + x2)(x2 + 3x + 1) Reorder the terms: 1 + -6x + 1x2 + 4x3 + x4 = (1 + 3x + x2)(1 + 3x + x2) Multiply (1 + 3x + x2) * (1 + 3x + x2) 1 + -6x + 1x2 + 4x3 + x4 = (1(1 + 3x + x2) + 3x * (1 + 3x + x2) + x2(1 + 3x + x2)) 1 + -6x + 1x2 + 4x3 + x4 = ((1 * 1 + 3x * 1 + x2 * 1) + 3x * (1 + 3x + x2) + x2(1 + 3x + x2)) 1 + -6x + 1x2 + 4x3 + x4 = ((1 + 3x + 1x2) + 3x * (1 + 3x + x2) + x2(1 + 3x + x2)) 1 + -6x + 1x2 + 4x3 + x4 = (1 + 3x + 1x2 + (1 * 3x + 3x * 3x + x2 * 3x) + x2(1 + 3x + x2)) 1 + -6x + 1x2 + 4x3 + x4 = (1 + 3x + 1x2 + (3x + 9x2 + 3x3) + x2(1 + 3x + x2)) 1 + -6x + 1x2 + 4x3 + x4 = (1 + 3x + 1x2 + 3x + 9x2 + 3x3 + (1 * x2 + 3x * x2 + x2 * x2)) 1 + -6x + 1x2 + 4x3 + x4 = (1 + 3x + 1x2 + 3x + 9x2 + 3x3 + (1x2 + 3x3 + x4)) Reorder the terms: 1 + -6x + 1x2 + 4x3 + x4 = (1 + 3x + 3x + 1x2 + 9x2 + 1x2 + 3x3 + 3x3 + x4) Combine like terms: 3x + 3x = 6x 1 + -6x + 1x2 + 4x3 + x4 = (1 + 6x + 1x2 + 9x2 + 1x2 + 3x3 + 3x3 + x4) Combine like terms: 1x2 + 9x2 = 10x2 1 + -6x + 1x2 + 4x3 + x4 = (1 + 6x + 10x2 + 1x2 + 3x3 + 3x3 + x4) Combine like terms: 10x2 + 1x2 = 11x2 1 + -6x + 1x2 + 4x3 + x4 = (1 + 6x + 11x2 + 3x3 + 3x3 + x4) Combine like terms: 3x3 + 3x3 = 6x3 1 + -6x + 1x2 + 4x3 + x4 = (1 + 6x + 11x2 + 6x3 + x4) Add '-1' to each side of the equation. 1 + -6x + 1x2 + 4x3 + -1 + x4 = 1 + 6x + 11x2 + 6x3 + -1 + x4 Reorder the terms: 1 + -1 + -6x + 1x2 + 4x3 + x4 = 1 + 6x + 11x2 + 6x3 + -1 + x4 Combine like terms: 1 + -1 = 0 0 + -6x + 1x2 + 4x3 + x4 = 1 + 6x + 11x2 + 6x3 + -1 + x4 -6x + 1x2 + 4x3 + x4 = 1 + 6x + 11x2 + 6x3 + -1 + x4 Reorder the terms: -6x + 1x2 + 4x3 + x4 = 1 + -1 + 6x + 11x2 + 6x3 + x4 Combine like terms: 1 + -1 = 0 -6x + 1x2 + 4x3 + x4 = 0 + 6x + 11x2 + 6x3 + x4 -6x + 1x2 + 4x3 + x4 = 6x + 11x2 + 6x3 + x4 Add '-1x4' to each side of the equation. -6x + 1x2 + 4x3 + x4 + -1x4 = 6x + 11x2 + 6x3 + x4 + -1x4 Combine like terms: x4 + -1x4 = 0 -6x + 1x2 + 4x3 + 0 = 6x + 11x2 + 6x3 + x4 + -1x4 -6x + 1x2 + 4x3 = 6x + 11x2 + 6x3 + x4 + -1x4 Combine like terms: x4 + -1x4 = 0 -6x + 1x2 + 4x3 = 6x + 11x2 + 6x3 + 0 -6x + 1x2 + 4x3 = 6x + 11x2 + 6x3 Solving -6x + 1x2 + 4x3 = 6x + 11x2 + 6x3 Solving for variable 'x'. Reorder the terms: -6x + -6x + 1x2 + -11x2 + 4x3 + -6x3 = 6x + 11x2 + 6x3 + -6x + -11x2 + -6x3 Combine like terms: -6x + -6x = -12x -12x + 1x2 + -11x2 + 4x3 + -6x3 = 6x + 11x2 + 6x3 + -6x + -11x2 + -6x3 Combine like terms: 1x2 + -11x2 = -10x2 -12x + -10x2 + 4x3 + -6x3 = 6x + 11x2 + 6x3 + -6x + -11x2 + -6x3 Combine like terms: 4x3 + -6x3 = -2x3 -12x + -10x2 + -2x3 = 6x + 11x2 + 6x3 + -6x + -11x2 + -6x3 Reorder the terms: -12x + -10x2 + -2x3 = 6x + -6x + 11x2 + -11x2 + 6x3 + -6x3 Combine like terms: 6x + -6x = 0 -12x + -10x2 + -2x3 = 0 + 11x2 + -11x2 + 6x3 + -6x3 -12x + -10x2 + -2x3 = 11x2 + -11x2 + 6x3 + -6x3 Combine like terms: 11x2 + -11x2 = 0 -12x + -10x2 + -2x3 = 0 + 6x3 + -6x3 -12x + -10x2 + -2x3 = 6x3 + -6x3 Combine like terms: 6x3 + -6x3 = 0 -12x + -10x2 + -2x3 = 0 Factor out the Greatest Common Factor (GCF), '-2x'. -2x(6 + 5x + x2) = 0 Factor a trinomial. -2x((3 + x)(2 + x)) = 0 Ignore the factor -2.Subproblem 1
Set the factor 'x' equal to zero and attempt to solve: Simplifying x = 0 Solving x = 0 Move all terms containing x to the left, all other terms to the right. Simplifying x = 0Subproblem 2
Set the factor '(3 + x)' equal to zero and attempt to solve: Simplifying 3 + x = 0 Solving 3 + x = 0 Move all terms containing x to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + x = 0 + -3 Combine like terms: 3 + -3 = 0 0 + x = 0 + -3 x = 0 + -3 Combine like terms: 0 + -3 = -3 x = -3 Simplifying x = -3Subproblem 3
Set the factor '(2 + x)' equal to zero and attempt to solve: Simplifying 2 + x = 0 Solving 2 + x = 0 Move all terms containing x to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + x = 0 + -2 Combine like terms: 2 + -2 = 0 0 + x = 0 + -2 x = 0 + -2 Combine like terms: 0 + -2 = -2 x = -2 Simplifying x = -2Solution
x = {0, -3, -2}
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