(x-7)(x-1)=(x+3)(x-4)+(x-4)(x-1)

Solution for (x-7)(x-1)=(x+3)(x-4)+(x-4)(x-1) equation:

Simplifying
(x + -7)(x + -1) = (x + 3)(x + -4) + (x + -4)(x + -1)

Reorder the terms:
(-7 + x)(x + -1) = (x + 3)(x + -4) + (x + -4)(x + -1)

Reorder the terms:
(-7 + x)(-1 + x) = (x + 3)(x + -4) + (x + -4)(x + -1)

Multiply (-7 + x) * (-1 + x)
(-7(-1 + x) + x(-1 + x)) = (x + 3)(x + -4) + (x + -4)(x + -1)
((-1 * -7 + x * -7) + x(-1 + x)) = (x + 3)(x + -4) + (x + -4)(x + -1)
((7 + -7x) + x(-1 + x)) = (x + 3)(x + -4) + (x + -4)(x + -1)
(7 + -7x + (-1 * x + x * x)) = (x + 3)(x + -4) + (x + -4)(x + -1)
(7 + -7x + (-1x + x2)) = (x + 3)(x + -4) + (x + -4)(x + -1)

Combine like terms: -7x + -1x = -8x
(7 + -8x + x2) = (x + 3)(x + -4) + (x + -4)(x + -1)

Reorder the terms:
7 + -8x + x2 = (3 + x)(x + -4) + (x + -4)(x + -1)

Reorder the terms:
7 + -8x + x2 = (3 + x)(-4 + x) + (x + -4)(x + -1)

Multiply (3 + x) * (-4 + x)
7 + -8x + x2 = (3(-4 + x) + x(-4 + x)) + (x + -4)(x + -1)
7 + -8x + x2 = ((-4 * 3 + x * 3) + x(-4 + x)) + (x + -4)(x + -1)
7 + -8x + x2 = ((-12 + 3x) + x(-4 + x)) + (x + -4)(x + -1)
7 + -8x + x2 = (-12 + 3x + (-4 * x + x * x)) + (x + -4)(x + -1)
7 + -8x + x2 = (-12 + 3x + (-4x + x2)) + (x + -4)(x + -1)

Combine like terms: 3x + -4x = -1x
7 + -8x + x2 = (-12 + -1x + x2) + (x + -4)(x + -1)

Reorder the terms:
7 + -8x + x2 = -12 + -1x + x2 + (-4 + x)(x + -1)

Reorder the terms:
7 + -8x + x2 = -12 + -1x + x2 + (-4 + x)(-1 + x)

Multiply (-4 + x) * (-1 + x)
7 + -8x + x2 = -12 + -1x + x2 + (-4(-1 + x) + x(-1 + x))
7 + -8x + x2 = -12 + -1x + x2 + ((-1 * -4 + x * -4) + x(-1 + x))
7 + -8x + x2 = -12 + -1x + x2 + ((4 + -4x) + x(-1 + x))
7 + -8x + x2 = -12 + -1x + x2 + (4 + -4x + (-1 * x + x * x))
7 + -8x + x2 = -12 + -1x + x2 + (4 + -4x + (-1x + x2))

Combine like terms: -4x + -1x = -5x
7 + -8x + x2 = -12 + -1x + x2 + (4 + -5x + x2)

Reorder the terms:
7 + -8x + x2 = -12 + 4 + -1x + -5x + x2 + x2

Combine like terms: -12 + 4 = -8
7 + -8x + x2 = -8 + -1x + -5x + x2 + x2

Combine like terms: -1x + -5x = -6x
7 + -8x + x2 = -8 + -6x + x2 + x2

Combine like terms: x2 + x2 = 2x2
7 + -8x + x2 = -8 + -6x + 2x2

Solving
7 + -8x + x2 = -8 + -6x + 2x2

Solving for variable 'x'.

Reorder the terms:
7 + 8 + -8x + 6x + x2 + -2x2 = -8 + -6x + 2x2 + 8 + 6x + -2x2

Combine like terms: 7 + 8 = 15
15 + -8x + 6x + x2 + -2x2 = -8 + -6x + 2x2 + 8 + 6x + -2x2

Combine like terms: -8x + 6x = -2x
15 + -2x + x2 + -2x2 = -8 + -6x + 2x2 + 8 + 6x + -2x2

Combine like terms: x2 + -2x2 = -1x2
15 + -2x + -1x2 = -8 + -6x + 2x2 + 8 + 6x + -2x2

Reorder the terms:
15 + -2x + -1x2 = -8 + 8 + -6x + 6x + 2x2 + -2x2

Combine like terms: -8 + 8 = 0
15 + -2x + -1x2 = 0 + -6x + 6x + 2x2 + -2x2
15 + -2x + -1x2 = -6x + 6x + 2x2 + -2x2

Combine like terms: -6x + 6x = 0
15 + -2x + -1x2 = 0 + 2x2 + -2x2
15 + -2x + -1x2 = 2x2 + -2x2

Combine like terms: 2x2 + -2x2 = 0
15 + -2x + -1x2 = 0

Factor a trinomial.
(3 + -1x)(5 + x) = 0

Subproblem 1
Set the factor '(3 + -1x)' equal to zero and attempt to solve:

Simplifying
3 + -1x = 0

Solving
3 + -1x = 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 + -1x = 0 + -3

Combine like terms: 3 + -3 = 0
0 + -1x = 0 + -3
-1x = 0 + -3

Combine like terms: 0 + -3 = -3
-1x = -3

Divide each side by '-1'.
x = 3

Simplifying
x = 3
Subproblem 2
Set the factor '(5 + x)' equal to zero and attempt to solve:

Simplifying
5 + x = 0

Solving
5 + x = 0

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

Add '-5' to each side of the equation.
5 + -5 + x = 0 + -5

Combine like terms: 5 + -5 = 0
0 + x = 0 + -5
x = 0 + -5

Combine like terms: 0 + -5 = -5
x = -5

Simplifying
x = -5
Solution
x = {3, -5}