y=y^2+4y+1

Solution for y=y^2+4y+1 equation:

Simplifying
y = y2 + 4y + 1

Reorder the terms:
y = 1 + 4y + y2

Solving
y = 1 + 4y + y2

Solving for variable 'y'.

Reorder the terms:
-1 + y + -4y + -1y2 = 1 + 4y + y2 + -1 + -4y + -1y2

Combine like terms: y + -4y = -3y
-1 + -3y + -1y2 = 1 + 4y + y2 + -1 + -4y + -1y2

Reorder the terms:
-1 + -3y + -1y2 = 1 + -1 + 4y + -4y + y2 + -1y2

Combine like terms: 1 + -1 = 0
-1 + -3y + -1y2 = 0 + 4y + -4y + y2 + -1y2
-1 + -3y + -1y2 = 4y + -4y + y2 + -1y2

Combine like terms: 4y + -4y = 0
-1 + -3y + -1y2 = 0 + y2 + -1y2
-1 + -3y + -1y2 = y2 + -1y2

Combine like terms: y2 + -1y2 = 0
-1 + -3y + -1y2 = 0

Factor out the Greatest Common Factor (GCF), '-1'.
-1(1 + 3y + y2) = 0

Ignore the factor -1.
Subproblem 1
Set the factor '(1 + 3y + y2)' equal to zero and attempt to solve:

Simplifying
1 + 3y + y2 = 0

Solving
1 + 3y + y2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + 3y + -1 + y2 = 0 + -1

Reorder the terms:
1 + -1 + 3y + y2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + 3y + y2 = 0 + -1
3y + y2 = 0 + -1

Combine like terms: 0 + -1 = -1
3y + y2 = -1

The y term is 3y.  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.
3y + 2.25 + y2 = -1 + 2.25

Reorder the terms:
2.25 + 3y + y2 = -1 + 2.25

Combine like terms: -1 + 2.25 = 1.25
2.25 + 3y + y2 = 1.25

Factor a perfect square on the left side:
(y + 1.5)(y + 1.5) = 1.25

Calculate the square root of the right side: 1.118033989

Break this problem into two subproblems by setting 
(y + 1.5) equal to 1.118033989 and -1.118033989.

Subproblem 1
y + 1.5 = 1.118033989

Simplifying
y + 1.5 = 1.118033989

Reorder the terms:
1.5 + y = 1.118033989

Solving
1.5 + y = 1.118033989

Solving for variable 'y'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + y = 1.118033989 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + y = 1.118033989 + -1.5
y = 1.118033989 + -1.5

Combine like terms: 1.118033989 + -1.5 = -0.381966011
y = -0.381966011

Simplifying
y = -0.381966011

Subproblem 2
y + 1.5 = -1.118033989

Simplifying
y + 1.5 = -1.118033989

Reorder the terms:
1.5 + y = -1.118033989

Solving
1.5 + y = -1.118033989

Solving for variable 'y'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + y = -1.118033989 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + y = -1.118033989 + -1.5
y = -1.118033989 + -1.5

Combine like terms: -1.118033989 + -1.5 = -2.618033989
y = -2.618033989

Simplifying
y = -2.618033989

Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {-0.381966011, -2.618033989}
Solution
y = {-0.381966011, -2.618033989}