4x(x+1)=75+x(x+7)

Solution for 4x(x+1)=75+x(x+7) equation:

Simplifying
4x(x + 1) = 75 + x(x + 7)

Reorder the terms:
4x(1 + x) = 75 + x(x + 7)
(1 * 4x + x * 4x) = 75 + x(x + 7)
(4x + 4x2) = 75 + x(x + 7)

Reorder the terms:
4x + 4x2 = 75 + x(7 + x)
4x + 4x2 = 75 + (7 * x + x * x)
4x + 4x2 = 75 + (7x + x2)

Solving
4x + 4x2 = 75 + 7x + x2

Solving for variable 'x'.

Reorder the terms:
-75 + 4x + -7x + 4x2 + -1x2 = 75 + 7x + x2 + -75 + -7x + -1x2

Combine like terms: 4x + -7x = -3x
-75 + -3x + 4x2 + -1x2 = 75 + 7x + x2 + -75 + -7x + -1x2

Combine like terms: 4x2 + -1x2 = 3x2
-75 + -3x + 3x2 = 75 + 7x + x2 + -75 + -7x + -1x2

Reorder the terms:
-75 + -3x + 3x2 = 75 + -75 + 7x + -7x + x2 + -1x2

Combine like terms: 75 + -75 = 0
-75 + -3x + 3x2 = 0 + 7x + -7x + x2 + -1x2
-75 + -3x + 3x2 = 7x + -7x + x2 + -1x2

Combine like terms: 7x + -7x = 0
-75 + -3x + 3x2 = 0 + x2 + -1x2
-75 + -3x + 3x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
-75 + -3x + 3x2 = 0

Factor out the Greatest Common Factor (GCF), '3'.
3(-25 + -1x + x2) = 0

Ignore the factor 3.
Subproblem 1
Set the factor '(-25 + -1x + x2)' equal to zero and attempt to solve:

Simplifying
-25 + -1x + x2 = 0

Solving
-25 + -1x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '25' to each side of the equation.
-25 + -1x + 25 + x2 = 0 + 25

Reorder the terms:
-25 + 25 + -1x + x2 = 0 + 25

Combine like terms: -25 + 25 = 0
0 + -1x + x2 = 0 + 25
-1x + x2 = 0 + 25

Combine like terms: 0 + 25 = 25
-1x + x2 = 25

The x term is -1x.  Take half its coefficient (-0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1x + 0.25 + x2 = 25 + 0.25

Reorder the terms:
0.25 + -1x + x2 = 25 + 0.25

Combine like terms: 25 + 0.25 = 25.25
0.25 + -1x + x2 = 25.25

Factor a perfect square on the left side:
(x + -0.5)(x + -0.5) = 25.25

Calculate the square root of the right side: 5.024937811

Break this problem into two subproblems by setting 
(x + -0.5) equal to 5.024937811 and -5.024937811.

Subproblem 1
x + -0.5 = 5.024937811

Simplifying
x + -0.5 = 5.024937811

Reorder the terms:
-0.5 + x = 5.024937811

Solving
-0.5 + x = 5.024937811

Solving for variable 'x'.

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

Add '0.5' to each side of the equation.
-0.5 + 0.5 + x = 5.024937811 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + x = 5.024937811 + 0.5
x = 5.024937811 + 0.5

Combine like terms: 5.024937811 + 0.5 = 5.524937811
x = 5.524937811

Simplifying
x = 5.524937811

Subproblem 2
x + -0.5 = -5.024937811

Simplifying
x + -0.5 = -5.024937811

Reorder the terms:
-0.5 + x = -5.024937811

Solving
-0.5 + x = -5.024937811

Solving for variable 'x'.

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

Add '0.5' to each side of the equation.
-0.5 + 0.5 + x = -5.024937811 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + x = -5.024937811 + 0.5
x = -5.024937811 + 0.5

Combine like terms: -5.024937811 + 0.5 = -4.524937811
x = -4.524937811

Simplifying
x = -4.524937811

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {5.524937811, -4.524937811}
Solution
x = {5.524937811, -4.524937811}