-2x(x+1)=7x+3

Solution for -2x(x+1)=7x+3 equation:

Simplifying
-2x(x + 1) = 7x + 3

Reorder the terms:
-2x(1 + x) = 7x + 3
(1 * -2x + x * -2x) = 7x + 3
(-2x + -2x2) = 7x + 3

Reorder the terms:
-2x + -2x2 = 3 + 7x

Solving
-2x + -2x2 = 3 + 7x

Solving for variable 'x'.

Reorder the terms:
-3 + -2x + -7x + -2x2 = 3 + 7x + -3 + -7x

Combine like terms: -2x + -7x = -9x
-3 + -9x + -2x2 = 3 + 7x + -3 + -7x

Reorder the terms:
-3 + -9x + -2x2 = 3 + -3 + 7x + -7x

Combine like terms: 3 + -3 = 0
-3 + -9x + -2x2 = 0 + 7x + -7x
-3 + -9x + -2x2 = 7x + -7x

Combine like terms: 7x + -7x = 0
-3 + -9x + -2x2 = 0

Factor out the Greatest Common Factor (GCF), '-1'.
-1(3 + 9x + 2x2) = 0

Ignore the factor -1.
Subproblem 1
Set the factor '(3 + 9x + 2x2)' equal to zero and attempt to solve:

Simplifying
3 + 9x + 2x2 = 0

Solving
3 + 9x + 2x2 = 0

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

Divide each side by '2'.
1.5 + 4.5x + x2 = 0

Move the constant term to the right:

Add '-1.5' to each side of the equation.
1.5 + 4.5x + -1.5 + x2 = 0 + -1.5

Reorder the terms:
1.5 + -1.5 + 4.5x + x2 = 0 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + 4.5x + x2 = 0 + -1.5
4.5x + x2 = 0 + -1.5

Combine like terms: 0 + -1.5 = -1.5
4.5x + x2 = -1.5

The x term is 4.5x.  Take half its coefficient (2.25).
Square it (5.0625) and add it to both sides.

Add '5.0625' to each side of the equation.
4.5x + 5.0625 + x2 = -1.5 + 5.0625

Reorder the terms:
5.0625 + 4.5x + x2 = -1.5 + 5.0625

Combine like terms: -1.5 + 5.0625 = 3.5625
5.0625 + 4.5x + x2 = 3.5625

Factor a perfect square on the left side:
(x + 2.25)(x + 2.25) = 3.5625

Calculate the square root of the right side: 1.887458609

Break this problem into two subproblems by setting 
(x + 2.25) equal to 1.887458609 and -1.887458609.

Subproblem 1
x + 2.25 = 1.887458609

Simplifying
x + 2.25 = 1.887458609

Reorder the terms:
2.25 + x = 1.887458609

Solving
2.25 + x = 1.887458609

Solving for variable 'x'.

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

Add '-2.25' to each side of the equation.
2.25 + -2.25 + x = 1.887458609 + -2.25

Combine like terms: 2.25 + -2.25 = 0.00
0.00 + x = 1.887458609 + -2.25
x = 1.887458609 + -2.25

Combine like terms: 1.887458609 + -2.25 = -0.362541391
x = -0.362541391

Simplifying
x = -0.362541391

Subproblem 2
x + 2.25 = -1.887458609

Simplifying
x + 2.25 = -1.887458609

Reorder the terms:
2.25 + x = -1.887458609

Solving
2.25 + x = -1.887458609

Solving for variable 'x'.

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

Add '-2.25' to each side of the equation.
2.25 + -2.25 + x = -1.887458609 + -2.25

Combine like terms: 2.25 + -2.25 = 0.00
0.00 + x = -1.887458609 + -2.25
x = -1.887458609 + -2.25

Combine like terms: -1.887458609 + -2.25 = -4.137458609
x = -4.137458609

Simplifying
x = -4.137458609

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.362541391, -4.137458609}
Solution
x = {-0.362541391, -4.137458609}