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

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

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

Reorder the terms:
a(-4 + a) = (a + 5) + -1(a + 1)
(-4 * a + a * a) = (a + 5) + -1(a + 1)
(-4a + a2) = (a + 5) + -1(a + 1)

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

Remove parenthesis around (5 + a)
-4a + a2 = 5 + a + -1(a + 1)

Reorder the terms:
-4a + a2 = 5 + a + -1(1 + a)
-4a + a2 = 5 + a + (1 * -1 + a * -1)
-4a + a2 = 5 + a + (-1 + -1a)

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

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

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

Solving
-4a + a2 = 4

Solving for variable 'a'.

Reorder the terms:
-4 + -4a + a2 = 4 + -4

Combine like terms: 4 + -4 = 0
-4 + -4a + a2 = 0

Begin completing the square.

Move the constant term to the right:

Add '4' to each side of the equation.
-4 + -4a + 4 + a2 = 0 + 4

Reorder the terms:
-4 + 4 + -4a + a2 = 0 + 4

Combine like terms: -4 + 4 = 0
0 + -4a + a2 = 0 + 4
-4a + a2 = 0 + 4

Combine like terms: 0 + 4 = 4
-4a + a2 = 4

The a term is -4a.  Take half its coefficient (-2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
-4a + 4 + a2 = 4 + 4

Reorder the terms:
4 + -4a + a2 = 4 + 4

Combine like terms: 4 + 4 = 8
4 + -4a + a2 = 8

Factor a perfect square on the left side:
(a + -2)(a + -2) = 8

Calculate the square root of the right side: 2.828427125

Break this problem into two subproblems by setting 
(a + -2) equal to 2.828427125 and -2.828427125.

Subproblem 1
a + -2 = 2.828427125

Simplifying
a + -2 = 2.828427125

Reorder the terms:
-2 + a = 2.828427125

Solving
-2 + a = 2.828427125

Solving for variable 'a'.

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

Add '2' to each side of the equation.
-2 + 2 + a = 2.828427125 + 2

Combine like terms: -2 + 2 = 0
0 + a = 2.828427125 + 2
a = 2.828427125 + 2

Combine like terms: 2.828427125 + 2 = 4.828427125
a = 4.828427125

Simplifying
a = 4.828427125

Subproblem 2
a + -2 = -2.828427125

Simplifying
a + -2 = -2.828427125

Reorder the terms:
-2 + a = -2.828427125

Solving
-2 + a = -2.828427125

Solving for variable 'a'.

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

Add '2' to each side of the equation.
-2 + 2 + a = -2.828427125 + 2

Combine like terms: -2 + 2 = 0
0 + a = -2.828427125 + 2
a = -2.828427125 + 2

Combine like terms: -2.828427125 + 2 = -0.828427125
a = -0.828427125

Simplifying
a = -0.828427125

Solution
The solution to the problem is based on the solutions
from the subproblems.
a = {4.828427125, -0.828427125}