5+4t(t-2)=2(t+7)+1

Solution for 5+4t(t-2)=2(t+7)+1 equation:

Simplifying
5 + 4t(t + -2) = 2(t + 7) + 1

Reorder the terms:
5 + 4t(-2 + t) = 2(t + 7) + 1
5 + (-2 * 4t + t * 4t) = 2(t + 7) + 1
5 + (-8t + 4t2) = 2(t + 7) + 1

Reorder the terms:
5 + -8t + 4t2 = 2(7 + t) + 1
5 + -8t + 4t2 = (7 * 2 + t * 2) + 1
5 + -8t + 4t2 = (14 + 2t) + 1

Reorder the terms:
5 + -8t + 4t2 = 14 + 1 + 2t

Combine like terms: 14 + 1 = 15
5 + -8t + 4t2 = 15 + 2t

Solving
5 + -8t + 4t2 = 15 + 2t

Solving for variable 't'.

Reorder the terms:
5 + -15 + -8t + -2t + 4t2 = 15 + 2t + -15 + -2t

Combine like terms: 5 + -15 = -10
-10 + -8t + -2t + 4t2 = 15 + 2t + -15 + -2t

Combine like terms: -8t + -2t = -10t
-10 + -10t + 4t2 = 15 + 2t + -15 + -2t

Reorder the terms:
-10 + -10t + 4t2 = 15 + -15 + 2t + -2t

Combine like terms: 15 + -15 = 0
-10 + -10t + 4t2 = 0 + 2t + -2t
-10 + -10t + 4t2 = 2t + -2t

Combine like terms: 2t + -2t = 0
-10 + -10t + 4t2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-5 + -5t + 2t2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-5 + -5t + 2t2)' equal to zero and attempt to solve:

Simplifying
-5 + -5t + 2t2 = 0

Solving
-5 + -5t + 2t2 = 0

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

Divide each side by '2'.
-2.5 + -2.5t + t2 = 0

Move the constant term to the right:

Add '2.5' to each side of the equation.
-2.5 + -2.5t + 2.5 + t2 = 0 + 2.5

Reorder the terms:
-2.5 + 2.5 + -2.5t + t2 = 0 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + -2.5t + t2 = 0 + 2.5
-2.5t + t2 = 0 + 2.5

Combine like terms: 0 + 2.5 = 2.5
-2.5t + t2 = 2.5

The t term is -2.5t.  Take half its coefficient (-1.25).
Square it (1.5625) and add it to both sides.

Add '1.5625' to each side of the equation.
-2.5t + 1.5625 + t2 = 2.5 + 1.5625

Reorder the terms:
1.5625 + -2.5t + t2 = 2.5 + 1.5625

Combine like terms: 2.5 + 1.5625 = 4.0625
1.5625 + -2.5t + t2 = 4.0625

Factor a perfect square on the left side:
(t + -1.25)(t + -1.25) = 4.0625

Calculate the square root of the right side: 2.015564437

Break this problem into two subproblems by setting 
(t + -1.25) equal to 2.015564437 and -2.015564437.

Subproblem 1
t + -1.25 = 2.015564437

Simplifying
t + -1.25 = 2.015564437

Reorder the terms:
-1.25 + t = 2.015564437

Solving
-1.25 + t = 2.015564437

Solving for variable 't'.

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

Add '1.25' to each side of the equation.
-1.25 + 1.25 + t = 2.015564437 + 1.25

Combine like terms: -1.25 + 1.25 = 0.00
0.00 + t = 2.015564437 + 1.25
t = 2.015564437 + 1.25

Combine like terms: 2.015564437 + 1.25 = 3.265564437
t = 3.265564437

Simplifying
t = 3.265564437

Subproblem 2
t + -1.25 = -2.015564437

Simplifying
t + -1.25 = -2.015564437

Reorder the terms:
-1.25 + t = -2.015564437

Solving
-1.25 + t = -2.015564437

Solving for variable 't'.

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

Add '1.25' to each side of the equation.
-1.25 + 1.25 + t = -2.015564437 + 1.25

Combine like terms: -1.25 + 1.25 = 0.00
0.00 + t = -2.015564437 + 1.25
t = -2.015564437 + 1.25

Combine like terms: -2.015564437 + 1.25 = -0.765564437
t = -0.765564437

Simplifying
t = -0.765564437

Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {3.265564437, -0.765564437}
Solution
t = {3.265564437, -0.765564437}