a+(a-4)=(a+5)-(a+1)a

Solution for a+(a-4)=(a+5)-(a+1)a equation:

Simplifying
a + (a + -4) = (a + 5) + -1(a + 1) * a

Reorder the terms:
a + (-4 + a) = (a + 5) + -1(a + 1) * a

Remove parenthesis around (-4 + a)
a + -4 + a = (a + 5) + -1(a + 1) * a

Reorder the terms:
-4 + a + a = (a + 5) + -1(a + 1) * a

Combine like terms: a + a = 2a
-4 + 2a = (a + 5) + -1(a + 1) * a

Reorder the terms:
-4 + 2a = (5 + a) + -1(a + 1) * a

Remove parenthesis around (5 + a)
-4 + 2a = 5 + a + -1(a + 1) * a

Reorder the terms:
-4 + 2a = 5 + a + -1(1 + a) * a

Reorder the terms for easier multiplication:
-4 + 2a = 5 + a + -1a(1 + a)
-4 + 2a = 5 + a + (1 * -1a + a * -1a)
-4 + 2a = 5 + a + (-1a + -1a2)

Combine like terms: a + -1a = 0
-4 + 2a = 5 + 0 + -1a2
-4 + 2a = 5 + -1a2

Solving
-4 + 2a = 5 + -1a2

Solving for variable 'a'.

Reorder the terms:
-4 + -5 + 2a + a2 = 5 + -1a2 + -5 + a2

Combine like terms: -4 + -5 = -9
-9 + 2a + a2 = 5 + -1a2 + -5 + a2

Reorder the terms:
-9 + 2a + a2 = 5 + -5 + -1a2 + a2

Combine like terms: 5 + -5 = 0
-9 + 2a + a2 = 0 + -1a2 + a2
-9 + 2a + a2 = -1a2 + a2

Combine like terms: -1a2 + a2 = 0
-9 + 2a + a2 = 0

Begin completing the square.

Move the constant term to the right:

Add '9' to each side of the equation.
-9 + 2a + 9 + a2 = 0 + 9

Reorder the terms:
-9 + 9 + 2a + a2 = 0 + 9

Combine like terms: -9 + 9 = 0
0 + 2a + a2 = 0 + 9
2a + a2 = 0 + 9

Combine like terms: 0 + 9 = 9
2a + a2 = 9

The a term is 2a.  Take half its coefficient (1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
2a + 1 + a2 = 9 + 1

Reorder the terms:
1 + 2a + a2 = 9 + 1

Combine like terms: 9 + 1 = 10
1 + 2a + a2 = 10

Factor a perfect square on the left side:
(a + 1)(a + 1) = 10

Calculate the square root of the right side: 3.16227766

Break this problem into two subproblems by setting 
(a + 1) equal to 3.16227766 and -3.16227766.

Subproblem 1
a + 1 = 3.16227766

Simplifying
a + 1 = 3.16227766

Reorder the terms:
1 + a = 3.16227766

Solving
1 + a = 3.16227766

Solving for variable 'a'.

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

Add '-1' to each side of the equation.
1 + -1 + a = 3.16227766 + -1

Combine like terms: 1 + -1 = 0
0 + a = 3.16227766 + -1
a = 3.16227766 + -1

Combine like terms: 3.16227766 + -1 = 2.16227766
a = 2.16227766

Simplifying
a = 2.16227766

Subproblem 2
a + 1 = -3.16227766

Simplifying
a + 1 = -3.16227766

Reorder the terms:
1 + a = -3.16227766

Solving
1 + a = -3.16227766

Solving for variable 'a'.

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

Add '-1' to each side of the equation.
1 + -1 + a = -3.16227766 + -1

Combine like terms: 1 + -1 = 0
0 + a = -3.16227766 + -1
a = -3.16227766 + -1

Combine like terms: -3.16227766 + -1 = -4.16227766
a = -4.16227766

Simplifying
a = -4.16227766

Solution
The solution to the problem is based on the solutions
from the subproblems.
a = {2.16227766, -4.16227766}