45+(x-3)(x+1)=3(x-4)(x+2)-2(x-3)(x-5)

Solution for 45+(x-3)(x+1)=3(x-4)(x+2)-2(x-3)(x-5) equation:

Simplifying
45 + (x + -3)(x + 1) = 3(x + -4)(x + 2) + -2(x + -3)(x + -5)

Reorder the terms:
45 + (-3 + x)(x + 1) = 3(x + -4)(x + 2) + -2(x + -3)(x + -5)

Reorder the terms:
45 + (-3 + x)(1 + x) = 3(x + -4)(x + 2) + -2(x + -3)(x + -5)

Multiply (-3 + x) * (1 + x)
45 + (-3(1 + x) + x(1 + x)) = 3(x + -4)(x + 2) + -2(x + -3)(x + -5)
45 + ((1 * -3 + x * -3) + x(1 + x)) = 3(x + -4)(x + 2) + -2(x + -3)(x + -5)
45 + ((-3 + -3x) + x(1 + x)) = 3(x + -4)(x + 2) + -2(x + -3)(x + -5)
45 + (-3 + -3x + (1 * x + x * x)) = 3(x + -4)(x + 2) + -2(x + -3)(x + -5)
45 + (-3 + -3x + (1x + x2)) = 3(x + -4)(x + 2) + -2(x + -3)(x + -5)

Combine like terms: -3x + 1x = -2x
45 + (-3 + -2x + x2) = 3(x + -4)(x + 2) + -2(x + -3)(x + -5)

Combine like terms: 45 + -3 = 42
42 + -2x + x2 = 3(x + -4)(x + 2) + -2(x + -3)(x + -5)

Reorder the terms:
42 + -2x + x2 = 3(-4 + x)(x + 2) + -2(x + -3)(x + -5)

Reorder the terms:
42 + -2x + x2 = 3(-4 + x)(2 + x) + -2(x + -3)(x + -5)

Multiply (-4 + x) * (2 + x)
42 + -2x + x2 = 3(-4(2 + x) + x(2 + x)) + -2(x + -3)(x + -5)
42 + -2x + x2 = 3((2 * -4 + x * -4) + x(2 + x)) + -2(x + -3)(x + -5)
42 + -2x + x2 = 3((-8 + -4x) + x(2 + x)) + -2(x + -3)(x + -5)
42 + -2x + x2 = 3(-8 + -4x + (2 * x + x * x)) + -2(x + -3)(x + -5)
42 + -2x + x2 = 3(-8 + -4x + (2x + x2)) + -2(x + -3)(x + -5)

Combine like terms: -4x + 2x = -2x
42 + -2x + x2 = 3(-8 + -2x + x2) + -2(x + -3)(x + -5)
42 + -2x + x2 = (-8 * 3 + -2x * 3 + x2 * 3) + -2(x + -3)(x + -5)
42 + -2x + x2 = (-24 + -6x + 3x2) + -2(x + -3)(x + -5)

Reorder the terms:
42 + -2x + x2 = -24 + -6x + 3x2 + -2(-3 + x)(x + -5)

Reorder the terms:
42 + -2x + x2 = -24 + -6x + 3x2 + -2(-3 + x)(-5 + x)

Multiply (-3 + x) * (-5 + x)
42 + -2x + x2 = -24 + -6x + 3x2 + -2(-3(-5 + x) + x(-5 + x))
42 + -2x + x2 = -24 + -6x + 3x2 + -2((-5 * -3 + x * -3) + x(-5 + x))
42 + -2x + x2 = -24 + -6x + 3x2 + -2((15 + -3x) + x(-5 + x))
42 + -2x + x2 = -24 + -6x + 3x2 + -2(15 + -3x + (-5 * x + x * x))
42 + -2x + x2 = -24 + -6x + 3x2 + -2(15 + -3x + (-5x + x2))

Combine like terms: -3x + -5x = -8x
42 + -2x + x2 = -24 + -6x + 3x2 + -2(15 + -8x + x2)
42 + -2x + x2 = -24 + -6x + 3x2 + (15 * -2 + -8x * -2 + x2 * -2)
42 + -2x + x2 = -24 + -6x + 3x2 + (-30 + 16x + -2x2)

Reorder the terms:
42 + -2x + x2 = -24 + -30 + -6x + 16x + 3x2 + -2x2

Combine like terms: -24 + -30 = -54
42 + -2x + x2 = -54 + -6x + 16x + 3x2 + -2x2

Combine like terms: -6x + 16x = 10x
42 + -2x + x2 = -54 + 10x + 3x2 + -2x2

Combine like terms: 3x2 + -2x2 = 1x2
42 + -2x + x2 = -54 + 10x + 1x2

Solving
42 + -2x + x2 = -54 + 10x + 1x2

Solving for variable 'x'.

Reorder the terms:
42 + 54 + -2x + -10x + x2 + -1x2 = -54 + 10x + 1x2 + 54 + -10x + -1x2

Combine like terms: 42 + 54 = 96
96 + -2x + -10x + x2 + -1x2 = -54 + 10x + 1x2 + 54 + -10x + -1x2

Combine like terms: -2x + -10x = -12x
96 + -12x + x2 + -1x2 = -54 + 10x + 1x2 + 54 + -10x + -1x2

Combine like terms: x2 + -1x2 = 0
96 + -12x + 0 = -54 + 10x + 1x2 + 54 + -10x + -1x2
96 + -12x = -54 + 10x + 1x2 + 54 + -10x + -1x2

Reorder the terms:
96 + -12x = -54 + 54 + 10x + -10x + 1x2 + -1x2

Combine like terms: -54 + 54 = 0
96 + -12x = 0 + 10x + -10x + 1x2 + -1x2
96 + -12x = 10x + -10x + 1x2 + -1x2

Combine like terms: 10x + -10x = 0
96 + -12x = 0 + 1x2 + -1x2
96 + -12x = 1x2 + -1x2

Combine like terms: 1x2 + -1x2 = 0
96 + -12x = 0

Factor out the Greatest Common Factor (GCF), '12'.
12(8 + -1x) = 0

Ignore the factor 12.
Subproblem 1
Set the factor '(8 + -1x)' equal to zero and attempt to solve:

Simplifying
8 + -1x = 0

Solving
8 + -1x = 0

Move all terms containing x to the left, all other terms to the right.

Add '-8' to each side of the equation.
8 + -8 + -1x = 0 + -8

Combine like terms: 8 + -8 = 0
0 + -1x = 0 + -8
-1x = 0 + -8

Combine like terms: 0 + -8 = -8
-1x = -8

Divide each side by '-1'.
x = 8

Simplifying
x = 8
Solution
x = {8}