a(a-3)=(a+4)-(a+1)

Solution for a(a-3)=(a+4)-(a+1) equation:

Simplifying
a(a + -3) = (a + 4) + -1(a + 1)

Reorder the terms:
a(-3 + a) = (a + 4) + -1(a + 1)
(-3 * a + a * a) = (a + 4) + -1(a + 1)
(-3a + a2) = (a + 4) + -1(a + 1)

Reorder the terms:
-3a + a2 = (4 + a) + -1(a + 1)

Remove parenthesis around (4 + a)
-3a + a2 = 4 + a + -1(a + 1)

Reorder the terms:
-3a + a2 = 4 + a + -1(1 + a)
-3a + a2 = 4 + a + (1 * -1 + a * -1)
-3a + a2 = 4 + a + (-1 + -1a)

Reorder the terms:
-3a + a2 = 4 + -1 + a + -1a

Combine like terms: 4 + -1 = 3
-3a + a2 = 3 + a + -1a

Combine like terms: a + -1a = 0
-3a + a2 = 3 + 0
-3a + a2 = 3

Solving
-3a + a2 = 3

Solving for variable 'a'.

Reorder the terms:
-3 + -3a + a2 = 3 + -3

Combine like terms: 3 + -3 = 0
-3 + -3a + a2 = 0

Begin completing the square.

Move the constant term to the right:

Add '3' to each side of the equation.
-3 + -3a + 3 + a2 = 0 + 3

Reorder the terms:
-3 + 3 + -3a + a2 = 0 + 3

Combine like terms: -3 + 3 = 0
0 + -3a + a2 = 0 + 3
-3a + a2 = 0 + 3

Combine like terms: 0 + 3 = 3
-3a + a2 = 3

The a term is -3a.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3a + 2.25 + a2 = 3 + 2.25

Reorder the terms:
2.25 + -3a + a2 = 3 + 2.25

Combine like terms: 3 + 2.25 = 5.25
2.25 + -3a + a2 = 5.25

Factor a perfect square on the left side:
(a + -1.5)(a + -1.5) = 5.25

Calculate the square root of the right side: 2.291287847

Break this problem into two subproblems by setting 
(a + -1.5) equal to 2.291287847 and -2.291287847.

Subproblem 1
a + -1.5 = 2.291287847

Simplifying
a + -1.5 = 2.291287847

Reorder the terms:
-1.5 + a = 2.291287847

Solving
-1.5 + a = 2.291287847

Solving for variable 'a'.

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

Add '1.5' to each side of the equation.
-1.5 + 1.5 + a = 2.291287847 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + a = 2.291287847 + 1.5
a = 2.291287847 + 1.5

Combine like terms: 2.291287847 + 1.5 = 3.791287847
a = 3.791287847

Simplifying
a = 3.791287847

Subproblem 2
a + -1.5 = -2.291287847

Simplifying
a + -1.5 = -2.291287847

Reorder the terms:
-1.5 + a = -2.291287847

Solving
-1.5 + a = -2.291287847

Solving for variable 'a'.

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

Add '1.5' to each side of the equation.
-1.5 + 1.5 + a = -2.291287847 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + a = -2.291287847 + 1.5
a = -2.291287847 + 1.5

Combine like terms: -2.291287847 + 1.5 = -0.791287847
a = -0.791287847

Simplifying
a = -0.791287847

Solution
The solution to the problem is based on the solutions
from the subproblems.
a = {3.791287847, -0.791287847}