3a+5a-a^2+10a+3+4a^2=

Solution for 3a+5a-a^2+10a+3+4a^2= equation:

Simplifying
3a + 5a + -1a2 + 10a + 3 + 4a2 = 0

Reorder the terms:
3 + 3a + 5a + 10a + -1a2 + 4a2 = 0

Combine like terms: 3a + 5a = 8a
3 + 8a + 10a + -1a2 + 4a2 = 0

Combine like terms: 8a + 10a = 18a
3 + 18a + -1a2 + 4a2 = 0

Combine like terms: -1a2 + 4a2 = 3a2
3 + 18a + 3a2 = 0

Solving
3 + 18a + 3a2 = 0

Solving for variable 'a'.

Factor out the Greatest Common Factor (GCF), '3'.
3(1 + 6a + a2) = 0

Ignore the factor 3.
Subproblem 1
Set the factor '(1 + 6a + a2)' equal to zero and attempt to solve:

Simplifying
1 + 6a + a2 = 0

Solving
1 + 6a + a2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + 6a + -1 + a2 = 0 + -1

Reorder the terms:
1 + -1 + 6a + a2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + 6a + a2 = 0 + -1
6a + a2 = 0 + -1

Combine like terms: 0 + -1 = -1
6a + a2 = -1

The a term is 6a.  Take half its coefficient (3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
6a + 9 + a2 = -1 + 9

Reorder the terms:
9 + 6a + a2 = -1 + 9

Combine like terms: -1 + 9 = 8
9 + 6a + a2 = 8

Factor a perfect square on the left side:
(a + 3)(a + 3) = 8

Calculate the square root of the right side: 2.828427125

Break this problem into two subproblems by setting 
(a + 3) equal to 2.828427125 and -2.828427125.

Subproblem 1
a + 3 = 2.828427125

Simplifying
a + 3 = 2.828427125

Reorder the terms:
3 + a = 2.828427125

Solving
3 + a = 2.828427125

Solving for variable 'a'.

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

Add '-3' to each side of the equation.
3 + -3 + a = 2.828427125 + -3

Combine like terms: 3 + -3 = 0
0 + a = 2.828427125 + -3
a = 2.828427125 + -3

Combine like terms: 2.828427125 + -3 = -0.171572875
a = -0.171572875

Simplifying
a = -0.171572875

Subproblem 2
a + 3 = -2.828427125

Simplifying
a + 3 = -2.828427125

Reorder the terms:
3 + a = -2.828427125

Solving
3 + a = -2.828427125

Solving for variable 'a'.

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

Add '-3' to each side of the equation.
3 + -3 + a = -2.828427125 + -3

Combine like terms: 3 + -3 = 0
0 + a = -2.828427125 + -3
a = -2.828427125 + -3

Combine like terms: -2.828427125 + -3 = -5.828427125
a = -5.828427125

Simplifying
a = -5.828427125

Solution
The solution to the problem is based on the solutions
from the subproblems.
a = {-0.171572875, -5.828427125}
Solution
a = {-0.171572875, -5.828427125}