5f+4f(f-1)=2+5(2+f)

Solution for 5f+4f(f-1)=2+5(2+f) equation:

Simplifying
5f + 4f(f + -1) = 2 + 5(2 + f)

Reorder the terms:
5f + 4f(-1 + f) = 2 + 5(2 + f)
5f + (-1 * 4f + f * 4f) = 2 + 5(2 + f)
5f + (-4f + 4f2) = 2 + 5(2 + f)

Combine like terms: 5f + -4f = 1f
1f + 4f2 = 2 + 5(2 + f)
1f + 4f2 = 2 + (2 * 5 + f * 5)
1f + 4f2 = 2 + (10 + 5f)

Combine like terms: 2 + 10 = 12
1f + 4f2 = 12 + 5f

Solving
1f + 4f2 = 12 + 5f

Solving for variable 'f'.

Reorder the terms:
-12 + 1f + -5f + 4f2 = 12 + 5f + -12 + -5f

Combine like terms: 1f + -5f = -4f
-12 + -4f + 4f2 = 12 + 5f + -12 + -5f

Reorder the terms:
-12 + -4f + 4f2 = 12 + -12 + 5f + -5f

Combine like terms: 12 + -12 = 0
-12 + -4f + 4f2 = 0 + 5f + -5f
-12 + -4f + 4f2 = 5f + -5f

Combine like terms: 5f + -5f = 0
-12 + -4f + 4f2 = 0

Factor out the Greatest Common Factor (GCF), '4'.
4(-3 + -1f + f2) = 0

Ignore the factor 4.
Subproblem 1
Set the factor '(-3 + -1f + f2)' equal to zero and attempt to solve:

Simplifying
-3 + -1f + f2 = 0

Solving
-3 + -1f + f2 = 0

Begin completing the square.

Move the constant term to the right:

Add '3' to each side of the equation.
-3 + -1f + 3 + f2 = 0 + 3

Reorder the terms:
-3 + 3 + -1f + f2 = 0 + 3

Combine like terms: -3 + 3 = 0
0 + -1f + f2 = 0 + 3
-1f + f2 = 0 + 3

Combine like terms: 0 + 3 = 3
-1f + f2 = 3

The f term is -1f.  Take half its coefficient (-0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1f + 0.25 + f2 = 3 + 0.25

Reorder the terms:
0.25 + -1f + f2 = 3 + 0.25

Combine like terms: 3 + 0.25 = 3.25
0.25 + -1f + f2 = 3.25

Factor a perfect square on the left side:
(f + -0.5)(f + -0.5) = 3.25

Calculate the square root of the right side: 1.802775638

Break this problem into two subproblems by setting 
(f + -0.5) equal to 1.802775638 and -1.802775638.

Subproblem 1
f + -0.5 = 1.802775638

Simplifying
f + -0.5 = 1.802775638

Reorder the terms:
-0.5 + f = 1.802775638

Solving
-0.5 + f = 1.802775638

Solving for variable 'f'.

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

Add '0.5' to each side of the equation.
-0.5 + 0.5 + f = 1.802775638 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + f = 1.802775638 + 0.5
f = 1.802775638 + 0.5

Combine like terms: 1.802775638 + 0.5 = 2.302775638
f = 2.302775638

Simplifying
f = 2.302775638

Subproblem 2
f + -0.5 = -1.802775638

Simplifying
f + -0.5 = -1.802775638

Reorder the terms:
-0.5 + f = -1.802775638

Solving
-0.5 + f = -1.802775638

Solving for variable 'f'.

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

Add '0.5' to each side of the equation.
-0.5 + 0.5 + f = -1.802775638 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + f = -1.802775638 + 0.5
f = -1.802775638 + 0.5

Combine like terms: -1.802775638 + 0.5 = -1.302775638
f = -1.302775638

Simplifying
f = -1.302775638

Solution
The solution to the problem is based on the solutions
from the subproblems.
f = {2.302775638, -1.302775638}
Solution
f = {2.302775638, -1.302775638}