3x-42=43(x+1)-12x(x+1)

Solution for 3x-42=43(x+1)-12x(x+1) equation:

Simplifying
3x + -42 = 43(x + 1) + -12x(x + 1)

Reorder the terms:
-42 + 3x = 43(x + 1) + -12x(x + 1)

Reorder the terms:
-42 + 3x = 43(1 + x) + -12x(x + 1)
-42 + 3x = (1 * 43 + x * 43) + -12x(x + 1)
-42 + 3x = (43 + 43x) + -12x(x + 1)

Reorder the terms:
-42 + 3x = 43 + 43x + -12x(1 + x)
-42 + 3x = 43 + 43x + (1 * -12x + x * -12x)
-42 + 3x = 43 + 43x + (-12x + -12x2)

Combine like terms: 43x + -12x = 31x
-42 + 3x = 43 + 31x + -12x2

Solving
-42 + 3x = 43 + 31x + -12x2

Solving for variable 'x'.

Reorder the terms:
-42 + -43 + 3x + -31x + 12x2 = 43 + 31x + -12x2 + -43 + -31x + 12x2

Combine like terms: -42 + -43 = -85
-85 + 3x + -31x + 12x2 = 43 + 31x + -12x2 + -43 + -31x + 12x2

Combine like terms: 3x + -31x = -28x
-85 + -28x + 12x2 = 43 + 31x + -12x2 + -43 + -31x + 12x2

Reorder the terms:
-85 + -28x + 12x2 = 43 + -43 + 31x + -31x + -12x2 + 12x2

Combine like terms: 43 + -43 = 0
-85 + -28x + 12x2 = 0 + 31x + -31x + -12x2 + 12x2
-85 + -28x + 12x2 = 31x + -31x + -12x2 + 12x2

Combine like terms: 31x + -31x = 0
-85 + -28x + 12x2 = 0 + -12x2 + 12x2
-85 + -28x + 12x2 = -12x2 + 12x2

Combine like terms: -12x2 + 12x2 = 0
-85 + -28x + 12x2 = 0

Begin completing the square.  Divide all terms by
12 the coefficient of the squared term: 

Divide each side by '12'.
-7.083333333 + -2.333333333x + x2 = 0

Move the constant term to the right:

Add '7.083333333' to each side of the equation.
-7.083333333 + -2.333333333x + 7.083333333 + x2 = 0 + 7.083333333

Reorder the terms:
-7.083333333 + 7.083333333 + -2.333333333x + x2 = 0 + 7.083333333

Combine like terms: -7.083333333 + 7.083333333 = 0.000000000
0.000000000 + -2.333333333x + x2 = 0 + 7.083333333
-2.333333333x + x2 = 0 + 7.083333333

Combine like terms: 0 + 7.083333333 = 7.083333333
-2.333333333x + x2 = 7.083333333

The x term is -2.333333333x.  Take half its coefficient (-1.166666667).
Square it (1.361111112) and add it to both sides.

Add '1.361111112' to each side of the equation.
-2.333333333x + 1.361111112 + x2 = 7.083333333 + 1.361111112

Reorder the terms:
1.361111112 + -2.333333333x + x2 = 7.083333333 + 1.361111112

Combine like terms: 7.083333333 + 1.361111112 = 8.444444445
1.361111112 + -2.333333333x + x2 = 8.444444445

Factor a perfect square on the left side:
(x + -1.166666667)(x + -1.166666667) = 8.444444445

Calculate the square root of the right side: 2.905932629

Break this problem into two subproblems by setting 
(x + -1.166666667) equal to 2.905932629 and -2.905932629.

Subproblem 1
x + -1.166666667 = 2.905932629

Simplifying
x + -1.166666667 = 2.905932629

Reorder the terms:
-1.166666667 + x = 2.905932629

Solving
-1.166666667 + x = 2.905932629

Solving for variable 'x'.

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

Add '1.166666667' to each side of the equation.
-1.166666667 + 1.166666667 + x = 2.905932629 + 1.166666667

Combine like terms: -1.166666667 + 1.166666667 = 0.000000000
0.000000000 + x = 2.905932629 + 1.166666667
x = 2.905932629 + 1.166666667

Combine like terms: 2.905932629 + 1.166666667 = 4.072599296
x = 4.072599296

Simplifying
x = 4.072599296

Subproblem 2
x + -1.166666667 = -2.905932629

Simplifying
x + -1.166666667 = -2.905932629

Reorder the terms:
-1.166666667 + x = -2.905932629

Solving
-1.166666667 + x = -2.905932629

Solving for variable 'x'.

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

Add '1.166666667' to each side of the equation.
-1.166666667 + 1.166666667 + x = -2.905932629 + 1.166666667

Combine like terms: -1.166666667 + 1.166666667 = 0.000000000
0.000000000 + x = -2.905932629 + 1.166666667
x = -2.905932629 + 1.166666667

Combine like terms: -2.905932629 + 1.166666667 = -1.739265962
x = -1.739265962

Simplifying
x = -1.739265962

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {4.072599296, -1.739265962}