(3t+5)(t+1)=(t+4)(t+2)

Solution for (3t+5)(t+1)=(t+4)(t+2) equation:

Simplifying
(3t + 5)(t + 1) = (t + 4)(t + 2)

Reorder the terms:
(5 + 3t)(t + 1) = (t + 4)(t + 2)

Reorder the terms:
(5 + 3t)(1 + t) = (t + 4)(t + 2)

Multiply (5 + 3t) * (1 + t)
(5(1 + t) + 3t * (1 + t)) = (t + 4)(t + 2)
((1 * 5 + t * 5) + 3t * (1 + t)) = (t + 4)(t + 2)
((5 + 5t) + 3t * (1 + t)) = (t + 4)(t + 2)
(5 + 5t + (1 * 3t + t * 3t)) = (t + 4)(t + 2)
(5 + 5t + (3t + 3t2)) = (t + 4)(t + 2)

Combine like terms: 5t + 3t = 8t
(5 + 8t + 3t2) = (t + 4)(t + 2)

Reorder the terms:
5 + 8t + 3t2 = (4 + t)(t + 2)

Reorder the terms:
5 + 8t + 3t2 = (4 + t)(2 + t)

Multiply (4 + t) * (2 + t)
5 + 8t + 3t2 = (4(2 + t) + t(2 + t))
5 + 8t + 3t2 = ((2 * 4 + t * 4) + t(2 + t))
5 + 8t + 3t2 = ((8 + 4t) + t(2 + t))
5 + 8t + 3t2 = (8 + 4t + (2 * t + t * t))
5 + 8t + 3t2 = (8 + 4t + (2t + t2))

Combine like terms: 4t + 2t = 6t
5 + 8t + 3t2 = (8 + 6t + t2)

Solving
5 + 8t + 3t2 = 8 + 6t + t2

Solving for variable 't'.

Reorder the terms:
5 + -8 + 8t + -6t + 3t2 + -1t2 = 8 + 6t + t2 + -8 + -6t + -1t2

Combine like terms: 5 + -8 = -3
-3 + 8t + -6t + 3t2 + -1t2 = 8 + 6t + t2 + -8 + -6t + -1t2

Combine like terms: 8t + -6t = 2t
-3 + 2t + 3t2 + -1t2 = 8 + 6t + t2 + -8 + -6t + -1t2

Combine like terms: 3t2 + -1t2 = 2t2
-3 + 2t + 2t2 = 8 + 6t + t2 + -8 + -6t + -1t2

Reorder the terms:
-3 + 2t + 2t2 = 8 + -8 + 6t + -6t + t2 + -1t2

Combine like terms: 8 + -8 = 0
-3 + 2t + 2t2 = 0 + 6t + -6t + t2 + -1t2
-3 + 2t + 2t2 = 6t + -6t + t2 + -1t2

Combine like terms: 6t + -6t = 0
-3 + 2t + 2t2 = 0 + t2 + -1t2
-3 + 2t + 2t2 = t2 + -1t2

Combine like terms: t2 + -1t2 = 0
-3 + 2t + 2t2 = 0

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

Divide each side by '2'.
-1.5 + t + t2 = 0

Move the constant term to the right:

Add '1.5' to each side of the equation.
-1.5 + t + 1.5 + t2 = 0 + 1.5

Reorder the terms:
-1.5 + 1.5 + t + t2 = 0 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + t + t2 = 0 + 1.5
t + t2 = 0 + 1.5

Combine like terms: 0 + 1.5 = 1.5
t + t2 = 1.5

The t term is t.  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.
 + 0.25 + t2 = 1.5 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + t2 = 1.5 + 0.25

Combine like terms: 1.5 + 0.25 = 1.75
1.25 + t2 = 1.75

Factor a perfect square on the left side:
(t + 0.5)(t + 0.5) = 1.75

Calculate the square root of the right side: 1.322875656

Break this problem into two subproblems by setting 
(t + 0.5) equal to 1.322875656 and -1.322875656.

Subproblem 1
t + 0.5 = 1.322875656

Simplifying
t + 0.5 = 1.322875656

Reorder the terms:
0.5 + t = 1.322875656

Solving
0.5 + t = 1.322875656

Solving for variable 't'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + t = 1.322875656 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + t = 1.322875656 + -0.5
t = 1.322875656 + -0.5

Combine like terms: 1.322875656 + -0.5 = 0.822875656
t = 0.822875656

Simplifying
t = 0.822875656

Subproblem 2
t + 0.5 = -1.322875656

Simplifying
t + 0.5 = -1.322875656

Reorder the terms:
0.5 + t = -1.322875656

Solving
0.5 + t = -1.322875656

Solving for variable 't'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + t = -1.322875656 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + t = -1.322875656 + -0.5
t = -1.322875656 + -0.5

Combine like terms: -1.322875656 + -0.5 = -1.822875656
t = -1.822875656

Simplifying
t = -1.822875656

Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {0.822875656, -1.822875656}