4x(x+7)=-6

Solution for 4x(x+7)=-6 equation:

Simplifying
4x(x + 7) = -6

Reorder the terms:
4x(7 + x) = -6
(7 * 4x + x * 4x) = -6
(28x + 4x2) = -6

Solving
28x + 4x2 = -6

Solving for variable 'x'.

Reorder the terms:
6 + 28x + 4x2 = -6 + 6

Combine like terms: -6 + 6 = 0
6 + 28x + 4x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(3 + 14x + 2x2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(3 + 14x + 2x2)' equal to zero and attempt to solve:

Simplifying
3 + 14x + 2x2 = 0

Solving
3 + 14x + 2x2 = 0

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

Divide each side by '2'.
1.5 + 7x + x2 = 0

Move the constant term to the right:

Add '-1.5' to each side of the equation.
1.5 + 7x + -1.5 + x2 = 0 + -1.5

Reorder the terms:
1.5 + -1.5 + 7x + x2 = 0 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + 7x + x2 = 0 + -1.5
7x + x2 = 0 + -1.5

Combine like terms: 0 + -1.5 = -1.5
7x + x2 = -1.5

The x term is 7x.  Take half its coefficient (3.5).
Square it (12.25) and add it to both sides.

Add '12.25' to each side of the equation.
7x + 12.25 + x2 = -1.5 + 12.25

Reorder the terms:
12.25 + 7x + x2 = -1.5 + 12.25

Combine like terms: -1.5 + 12.25 = 10.75
12.25 + 7x + x2 = 10.75

Factor a perfect square on the left side:
(x + 3.5)(x + 3.5) = 10.75

Calculate the square root of the right side: 3.278719262

Break this problem into two subproblems by setting 
(x + 3.5) equal to 3.278719262 and -3.278719262.

Subproblem 1
x + 3.5 = 3.278719262

Simplifying
x + 3.5 = 3.278719262

Reorder the terms:
3.5 + x = 3.278719262

Solving
3.5 + x = 3.278719262

Solving for variable 'x'.

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

Add '-3.5' to each side of the equation.
3.5 + -3.5 + x = 3.278719262 + -3.5

Combine like terms: 3.5 + -3.5 = 0.0
0.0 + x = 3.278719262 + -3.5
x = 3.278719262 + -3.5

Combine like terms: 3.278719262 + -3.5 = -0.221280738
x = -0.221280738

Simplifying
x = -0.221280738

Subproblem 2
x + 3.5 = -3.278719262

Simplifying
x + 3.5 = -3.278719262

Reorder the terms:
3.5 + x = -3.278719262

Solving
3.5 + x = -3.278719262

Solving for variable 'x'.

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

Add '-3.5' to each side of the equation.
3.5 + -3.5 + x = -3.278719262 + -3.5

Combine like terms: 3.5 + -3.5 = 0.0
0.0 + x = -3.278719262 + -3.5
x = -3.278719262 + -3.5

Combine like terms: -3.278719262 + -3.5 = -6.778719262
x = -6.778719262

Simplifying
x = -6.778719262

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.221280738, -6.778719262}
Solution
x = {-0.221280738, -6.778719262}