x(x+1(x+2(x+3(x+4))))=10

Solution for x(x+1(x+2(x+3(x+4))))=10 equation:

Simplifying
x(x + 1(x + 2(x + 3(x + 4)))) = 10

Reorder the terms:
x(x + 1(x + 2(x + 3(4 + x)))) = 10
x(x + 1(x + 2(x + (4 * 3 + x * 3)))) = 10
x(x + 1(x + 2(x + (12 + 3x)))) = 10

Reorder the terms:
x(x + 1(x + 2(12 + x + 3x))) = 10

Combine like terms: x + 3x = 4x
x(x + 1(x + 2(12 + 4x))) = 10
x(x + 1(x + (12 * 2 + 4x * 2))) = 10
x(x + 1(x + (24 + 8x))) = 10

Reorder the terms:
x(x + 1(24 + x + 8x)) = 10

Combine like terms: x + 8x = 9x
x(x + 1(24 + 9x)) = 10
x(x + (24 * 1 + 9x * 1)) = 10
x(x + (24 + 9x)) = 10

Reorder the terms:
x(24 + x + 9x) = 10

Combine like terms: x + 9x = 10x
x(24 + 10x) = 10
(24 * x + 10x * x) = 10
(24x + 10x2) = 10

Solving
24x + 10x2 = 10

Solving for variable 'x'.

Reorder the terms:
-10 + 24x + 10x2 = 10 + -10

Combine like terms: 10 + -10 = 0
-10 + 24x + 10x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-5 + 12x + 5x2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-5 + 12x + 5x2)' equal to zero and attempt to solve:

Simplifying
-5 + 12x + 5x2 = 0

Solving
-5 + 12x + 5x2 = 0

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

Divide each side by '5'.
-1 + 2.4x + x2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + 2.4x + 1 + x2 = 0 + 1

Reorder the terms:
-1 + 1 + 2.4x + x2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + 2.4x + x2 = 0 + 1
2.4x + x2 = 0 + 1

Combine like terms: 0 + 1 = 1
2.4x + x2 = 1

The x term is 2.4x.  Take half its coefficient (1.2).
Square it (1.44) and add it to both sides.

Add '1.44' to each side of the equation.
2.4x + 1.44 + x2 = 1 + 1.44

Reorder the terms:
1.44 + 2.4x + x2 = 1 + 1.44

Combine like terms: 1 + 1.44 = 2.44
1.44 + 2.4x + x2 = 2.44

Factor a perfect square on the left side:
(x + 1.2)(x + 1.2) = 2.44

Calculate the square root of the right side: 1.562049935

Break this problem into two subproblems by setting 
(x + 1.2) equal to 1.562049935 and -1.562049935.

Subproblem 1
x + 1.2 = 1.562049935

Simplifying
x + 1.2 = 1.562049935

Reorder the terms:
1.2 + x = 1.562049935

Solving
1.2 + x = 1.562049935

Solving for variable 'x'.

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

Add '-1.2' to each side of the equation.
1.2 + -1.2 + x = 1.562049935 + -1.2

Combine like terms: 1.2 + -1.2 = 0.0
0.0 + x = 1.562049935 + -1.2
x = 1.562049935 + -1.2

Combine like terms: 1.562049935 + -1.2 = 0.362049935
x = 0.362049935

Simplifying
x = 0.362049935

Subproblem 2
x + 1.2 = -1.562049935

Simplifying
x + 1.2 = -1.562049935

Reorder the terms:
1.2 + x = -1.562049935

Solving
1.2 + x = -1.562049935

Solving for variable 'x'.

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

Add '-1.2' to each side of the equation.
1.2 + -1.2 + x = -1.562049935 + -1.2

Combine like terms: 1.2 + -1.2 = 0.0
0.0 + x = -1.562049935 + -1.2
x = -1.562049935 + -1.2

Combine like terms: -1.562049935 + -1.2 = -2.762049935
x = -2.762049935

Simplifying
x = -2.762049935

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.362049935, -2.762049935}
Solution
x = {0.362049935, -2.762049935}