x(x+15)=x+45(x+40)x+25

Solution for x(x+15)=x+45(x+40)x+25 equation:

Simplifying
x(x + 15) = x + 45(x + 40) * x + 25

Reorder the terms:
x(15 + x) = x + 45(x + 40) * x + 25
(15 * x + x * x) = x + 45(x + 40) * x + 25
(15x + x2) = x + 45(x + 40) * x + 25

Reorder the terms:
15x + x2 = x + 45(40 + x) * x + 25

Reorder the terms for easier multiplication:
15x + x2 = x + 45x(40 + x) + 25
15x + x2 = x + (40 * 45x + x * 45x) + 25
15x + x2 = x + (1800x + 45x2) + 25

Reorder the terms:
15x + x2 = 25 + x + 1800x + 45x2

Combine like terms: x + 1800x = 1801x
15x + x2 = 25 + 1801x + 45x2

Solving
15x + x2 = 25 + 1801x + 45x2

Solving for variable 'x'.

Reorder the terms:
-25 + 15x + -1801x + x2 + -45x2 = 25 + 1801x + 45x2 + -25 + -1801x + -45x2

Combine like terms: 15x + -1801x = -1786x
-25 + -1786x + x2 + -45x2 = 25 + 1801x + 45x2 + -25 + -1801x + -45x2

Combine like terms: x2 + -45x2 = -44x2
-25 + -1786x + -44x2 = 25 + 1801x + 45x2 + -25 + -1801x + -45x2

Reorder the terms:
-25 + -1786x + -44x2 = 25 + -25 + 1801x + -1801x + 45x2 + -45x2

Combine like terms: 25 + -25 = 0
-25 + -1786x + -44x2 = 0 + 1801x + -1801x + 45x2 + -45x2
-25 + -1786x + -44x2 = 1801x + -1801x + 45x2 + -45x2

Combine like terms: 1801x + -1801x = 0
-25 + -1786x + -44x2 = 0 + 45x2 + -45x2
-25 + -1786x + -44x2 = 45x2 + -45x2

Combine like terms: 45x2 + -45x2 = 0
-25 + -1786x + -44x2 = 0

Factor out the Greatest Common Factor (GCF), '-1'.
-1(25 + 1786x + 44x2) = 0

Ignore the factor -1.
Subproblem 1
Set the factor '(25 + 1786x + 44x2)' equal to zero and attempt to solve:

Simplifying
25 + 1786x + 44x2 = 0

Solving
25 + 1786x + 44x2 = 0

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

Divide each side by '44'.
0.5681818182 + 40.59090909x + x2 = 0

Move the constant term to the right:

Add '-0.5681818182' to each side of the equation.
0.5681818182 + 40.59090909x + -0.5681818182 + x2 = 0 + -0.5681818182

Reorder the terms:
0.5681818182 + -0.5681818182 + 40.59090909x + x2 = 0 + -0.5681818182

Combine like terms: 0.5681818182 + -0.5681818182 = 0.0000000000
0.0000000000 + 40.59090909x + x2 = 0 + -0.5681818182
40.59090909x + x2 = 0 + -0.5681818182

Combine like terms: 0 + -0.5681818182 = -0.5681818182
40.59090909x + x2 = -0.5681818182

The x term is 40.59090909x.  Take half its coefficient (20.29545455).
Square it (411.9054754) and add it to both sides.

Add '411.9054754' to each side of the equation.
40.59090909x + 411.9054754 + x2 = -0.5681818182 + 411.9054754

Reorder the terms:
411.9054754 + 40.59090909x + x2 = -0.5681818182 + 411.9054754

Combine like terms: -0.5681818182 + 411.9054754 = 411.3372935818
411.9054754 + 40.59090909x + x2 = 411.3372935818

Factor a perfect square on the left side:
(x + 20.29545455)(x + 20.29545455) = 411.3372935818

Calculate the square root of the right side: 20.281451959

Break this problem into two subproblems by setting 
(x + 20.29545455) equal to 20.281451959 and -20.281451959.

Subproblem 1
x + 20.29545455 = 20.281451959

Simplifying
x + 20.29545455 = 20.281451959

Reorder the terms:
20.29545455 + x = 20.281451959

Solving
20.29545455 + x = 20.281451959

Solving for variable 'x'.

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

Add '-20.29545455' to each side of the equation.
20.29545455 + -20.29545455 + x = 20.281451959 + -20.29545455

Combine like terms: 20.29545455 + -20.29545455 = 0.00000000
0.00000000 + x = 20.281451959 + -20.29545455
x = 20.281451959 + -20.29545455

Combine like terms: 20.281451959 + -20.29545455 = -0.014002591
x = -0.014002591

Simplifying
x = -0.014002591

Subproblem 2
x + 20.29545455 = -20.281451959

Simplifying
x + 20.29545455 = -20.281451959

Reorder the terms:
20.29545455 + x = -20.281451959

Solving
20.29545455 + x = -20.281451959

Solving for variable 'x'.

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

Add '-20.29545455' to each side of the equation.
20.29545455 + -20.29545455 + x = -20.281451959 + -20.29545455

Combine like terms: 20.29545455 + -20.29545455 = 0.00000000
0.00000000 + x = -20.281451959 + -20.29545455
x = -20.281451959 + -20.29545455

Combine like terms: -20.281451959 + -20.29545455 = -40.576906509
x = -40.576906509

Simplifying
x = -40.576906509

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.014002591, -40.576906509}
Solution
x = {-0.014002591, -40.576906509}