(3x+1)(x+3)=19-2(x+2)

Solution for (3x+1)(x+3)=19-2(x+2) equation:

Simplifying
(3x + 1)(x + 3) = 19 + -2(x + 2)

Reorder the terms:
(1 + 3x)(x + 3) = 19 + -2(x + 2)

Reorder the terms:
(1 + 3x)(3 + x) = 19 + -2(x + 2)

Multiply (1 + 3x) * (3 + x)
(1(3 + x) + 3x * (3 + x)) = 19 + -2(x + 2)
((3 * 1 + x * 1) + 3x * (3 + x)) = 19 + -2(x + 2)
((3 + 1x) + 3x * (3 + x)) = 19 + -2(x + 2)
(3 + 1x + (3 * 3x + x * 3x)) = 19 + -2(x + 2)
(3 + 1x + (9x + 3x2)) = 19 + -2(x + 2)

Combine like terms: 1x + 9x = 10x
(3 + 10x + 3x2) = 19 + -2(x + 2)

Reorder the terms:
3 + 10x + 3x2 = 19 + -2(2 + x)
3 + 10x + 3x2 = 19 + (2 * -2 + x * -2)
3 + 10x + 3x2 = 19 + (-4 + -2x)

Combine like terms: 19 + -4 = 15
3 + 10x + 3x2 = 15 + -2x

Solving
3 + 10x + 3x2 = 15 + -2x

Solving for variable 'x'.

Reorder the terms:
3 + -15 + 10x + 2x + 3x2 = 15 + -2x + -15 + 2x

Combine like terms: 3 + -15 = -12
-12 + 10x + 2x + 3x2 = 15 + -2x + -15 + 2x

Combine like terms: 10x + 2x = 12x
-12 + 12x + 3x2 = 15 + -2x + -15 + 2x

Reorder the terms:
-12 + 12x + 3x2 = 15 + -15 + -2x + 2x

Combine like terms: 15 + -15 = 0
-12 + 12x + 3x2 = 0 + -2x + 2x
-12 + 12x + 3x2 = -2x + 2x

Combine like terms: -2x + 2x = 0
-12 + 12x + 3x2 = 0

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

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

Simplifying
-4 + 4x + x2 = 0

Solving
-4 + 4x + x2 = 0

Begin completing the square.

Move the constant term to the right:

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

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

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

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

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 = 4 + 4

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

Combine like terms: 4 + 4 = 8
4 + 4x + x2 = 8

Factor a perfect square on the left side:
(x + 2)(x + 2) = 8

Calculate the square root of the right side: 2.828427125

Break this problem into two subproblems by setting 
(x + 2) equal to 2.828427125 and -2.828427125.

Subproblem 1
x + 2 = 2.828427125

Simplifying
x + 2 = 2.828427125

Reorder the terms:
2 + x = 2.828427125

Solving
2 + x = 2.828427125

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 = 2.828427125 + -2

Combine like terms: 2 + -2 = 0
0 + x = 2.828427125 + -2
x = 2.828427125 + -2

Combine like terms: 2.828427125 + -2 = 0.828427125
x = 0.828427125

Simplifying
x = 0.828427125

Subproblem 2
x + 2 = -2.828427125

Simplifying
x + 2 = -2.828427125

Reorder the terms:
2 + x = -2.828427125

Solving
2 + x = -2.828427125

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 = -2.828427125 + -2

Combine like terms: 2 + -2 = 0
0 + x = -2.828427125 + -2
x = -2.828427125 + -2

Combine like terms: -2.828427125 + -2 = -4.828427125
x = -4.828427125

Simplifying
x = -4.828427125

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.828427125, -4.828427125}
Solution
x = {0.828427125, -4.828427125}