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

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

Simplifying
3x + 1 + 2(1 + 3x2) = 5(1 + -1x) + 4(x2 + 1)
3x + 1 + (1 * 2 + 3x2 * 2) = 5(1 + -1x) + 4(x2 + 1)
3x + 1 + (2 + 6x2) = 5(1 + -1x) + 4(x2 + 1)

Reorder the terms:
1 + 2 + 3x + 6x2 = 5(1 + -1x) + 4(x2 + 1)

Combine like terms: 1 + 2 = 3
3 + 3x + 6x2 = 5(1 + -1x) + 4(x2 + 1)
3 + 3x + 6x2 = (1 * 5 + -1x * 5) + 4(x2 + 1)
3 + 3x + 6x2 = (5 + -5x) + 4(x2 + 1)

Reorder the terms:
3 + 3x + 6x2 = 5 + -5x + 4(1 + x2)
3 + 3x + 6x2 = 5 + -5x + (1 * 4 + x2 * 4)
3 + 3x + 6x2 = 5 + -5x + (4 + 4x2)

Reorder the terms:
3 + 3x + 6x2 = 5 + 4 + -5x + 4x2

Combine like terms: 5 + 4 = 9
3 + 3x + 6x2 = 9 + -5x + 4x2

Solving
3 + 3x + 6x2 = 9 + -5x + 4x2

Solving for variable 'x'.

Reorder the terms:
3 + -9 + 3x + 5x + 6x2 + -4x2 = 9 + -5x + 4x2 + -9 + 5x + -4x2

Combine like terms: 3 + -9 = -6
-6 + 3x + 5x + 6x2 + -4x2 = 9 + -5x + 4x2 + -9 + 5x + -4x2

Combine like terms: 3x + 5x = 8x
-6 + 8x + 6x2 + -4x2 = 9 + -5x + 4x2 + -9 + 5x + -4x2

Combine like terms: 6x2 + -4x2 = 2x2
-6 + 8x + 2x2 = 9 + -5x + 4x2 + -9 + 5x + -4x2

Reorder the terms:
-6 + 8x + 2x2 = 9 + -9 + -5x + 5x + 4x2 + -4x2

Combine like terms: 9 + -9 = 0
-6 + 8x + 2x2 = 0 + -5x + 5x + 4x2 + -4x2
-6 + 8x + 2x2 = -5x + 5x + 4x2 + -4x2

Combine like terms: -5x + 5x = 0
-6 + 8x + 2x2 = 0 + 4x2 + -4x2
-6 + 8x + 2x2 = 4x2 + -4x2

Combine like terms: 4x2 + -4x2 = 0
-6 + 8x + 2x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-3 + 4x + x2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-3 + 4x + x2)' equal to zero and attempt to solve:

Simplifying
-3 + 4x + x2 = 0

Solving
-3 + 4x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '3' to each side of the equation.
-3 + 4x + 3 + x2 = 0 + 3

Reorder the terms:
-3 + 3 + 4x + x2 = 0 + 3

Combine like terms: -3 + 3 = 0
0 + 4x + x2 = 0 + 3
4x + x2 = 0 + 3

Combine like terms: 0 + 3 = 3
4x + x2 = 3

The x term is 4x.  Take half its coefficient (2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
4x + 4 + x2 = 3 + 4

Reorder the terms:
4 + 4x + x2 = 3 + 4

Combine like terms: 3 + 4 = 7
4 + 4x + x2 = 7

Factor a perfect square on the left side:
(x + 2)(x + 2) = 7

Calculate the square root of the right side: 2.645751311

Break this problem into two subproblems by setting 
(x + 2) equal to 2.645751311 and -2.645751311.

Subproblem 1
x + 2 = 2.645751311

Simplifying
x + 2 = 2.645751311

Reorder the terms:
2 + x = 2.645751311

Solving
2 + x = 2.645751311

Solving for variable 'x'.

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 = 2.645751311 + -2

Combine like terms: 2 + -2 = 0
0 + x = 2.645751311 + -2
x = 2.645751311 + -2

Combine like terms: 2.645751311 + -2 = 0.645751311
x = 0.645751311

Simplifying
x = 0.645751311

Subproblem 2
x + 2 = -2.645751311

Simplifying
x + 2 = -2.645751311

Reorder the terms:
2 + x = -2.645751311

Solving
2 + x = -2.645751311

Solving for variable 'x'.

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 = -2.645751311 + -2

Combine like terms: 2 + -2 = 0
0 + x = -2.645751311 + -2
x = -2.645751311 + -2

Combine like terms: -2.645751311 + -2 = -4.645751311
x = -4.645751311

Simplifying
x = -4.645751311

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.645751311, -4.645751311}
Solution
x = {0.645751311, -4.645751311}