(a+2)(2a+5)=2(a+1)+13

Solution for (a+2)(2a+5)=2(a+1)+13 equation:

Simplifying
(a + 2)(2a + 5) = 2(a + 1) + 13

Reorder the terms:
(2 + a)(2a + 5) = 2(a + 1) + 13

Reorder the terms:
(2 + a)(5 + 2a) = 2(a + 1) + 13

Multiply (2 + a) * (5 + 2a)
(2(5 + 2a) + a(5 + 2a)) = 2(a + 1) + 13
((5 * 2 + 2a * 2) + a(5 + 2a)) = 2(a + 1) + 13
((10 + 4a) + a(5 + 2a)) = 2(a + 1) + 13
(10 + 4a + (5 * a + 2a * a)) = 2(a + 1) + 13
(10 + 4a + (5a + 2a2)) = 2(a + 1) + 13

Combine like terms: 4a + 5a = 9a
(10 + 9a + 2a2) = 2(a + 1) + 13

Reorder the terms:
10 + 9a + 2a2 = 2(1 + a) + 13
10 + 9a + 2a2 = (1 * 2 + a * 2) + 13
10 + 9a + 2a2 = (2 + 2a) + 13

Reorder the terms:
10 + 9a + 2a2 = 2 + 13 + 2a

Combine like terms: 2 + 13 = 15
10 + 9a + 2a2 = 15 + 2a

Solving
10 + 9a + 2a2 = 15 + 2a

Solving for variable 'a'.

Reorder the terms:
10 + -15 + 9a + -2a + 2a2 = 15 + 2a + -15 + -2a

Combine like terms: 10 + -15 = -5
-5 + 9a + -2a + 2a2 = 15 + 2a + -15 + -2a

Combine like terms: 9a + -2a = 7a
-5 + 7a + 2a2 = 15 + 2a + -15 + -2a

Reorder the terms:
-5 + 7a + 2a2 = 15 + -15 + 2a + -2a

Combine like terms: 15 + -15 = 0
-5 + 7a + 2a2 = 0 + 2a + -2a
-5 + 7a + 2a2 = 2a + -2a

Combine like terms: 2a + -2a = 0
-5 + 7a + 2a2 = 0

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

Divide each side by '2'.
-2.5 + 3.5a + a2 = 0

Move the constant term to the right:

Add '2.5' to each side of the equation.
-2.5 + 3.5a + 2.5 + a2 = 0 + 2.5

Reorder the terms:
-2.5 + 2.5 + 3.5a + a2 = 0 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + 3.5a + a2 = 0 + 2.5
3.5a + a2 = 0 + 2.5

Combine like terms: 0 + 2.5 = 2.5
3.5a + a2 = 2.5

The a term is 3.5a.  Take half its coefficient (1.75).
Square it (3.0625) and add it to both sides.

Add '3.0625' to each side of the equation.
3.5a + 3.0625 + a2 = 2.5 + 3.0625

Reorder the terms:
3.0625 + 3.5a + a2 = 2.5 + 3.0625

Combine like terms: 2.5 + 3.0625 = 5.5625
3.0625 + 3.5a + a2 = 5.5625

Factor a perfect square on the left side:
(a + 1.75)(a + 1.75) = 5.5625

Calculate the square root of the right side: 2.358495283

Break this problem into two subproblems by setting 
(a + 1.75) equal to 2.358495283 and -2.358495283.

Subproblem 1
a + 1.75 = 2.358495283

Simplifying
a + 1.75 = 2.358495283

Reorder the terms:
1.75 + a = 2.358495283

Solving
1.75 + a = 2.358495283

Solving for variable 'a'.

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

Add '-1.75' to each side of the equation.
1.75 + -1.75 + a = 2.358495283 + -1.75

Combine like terms: 1.75 + -1.75 = 0.00
0.00 + a = 2.358495283 + -1.75
a = 2.358495283 + -1.75

Combine like terms: 2.358495283 + -1.75 = 0.608495283
a = 0.608495283

Simplifying
a = 0.608495283

Subproblem 2
a + 1.75 = -2.358495283

Simplifying
a + 1.75 = -2.358495283

Reorder the terms:
1.75 + a = -2.358495283

Solving
1.75 + a = -2.358495283

Solving for variable 'a'.

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

Add '-1.75' to each side of the equation.
1.75 + -1.75 + a = -2.358495283 + -1.75

Combine like terms: 1.75 + -1.75 = 0.00
0.00 + a = -2.358495283 + -1.75
a = -2.358495283 + -1.75

Combine like terms: -2.358495283 + -1.75 = -4.108495283
a = -4.108495283

Simplifying
a = -4.108495283

Solution
The solution to the problem is based on the solutions
from the subproblems.
a = {0.608495283, -4.108495283}