2(3t-4)-(t^2+2)-5t(t-3)=0

Solution for 2(3t-4)-(t^2+2)-5t(t-3)=0 equation:

Simplifying
2(3t + -4) + -1(t2 + 2) + -5t(t + -3) = 0

Reorder the terms:
2(-4 + 3t) + -1(t2 + 2) + -5t(t + -3) = 0
(-4 * 2 + 3t * 2) + -1(t2 + 2) + -5t(t + -3) = 0
(-8 + 6t) + -1(t2 + 2) + -5t(t + -3) = 0

Reorder the terms:
-8 + 6t + -1(2 + t2) + -5t(t + -3) = 0
-8 + 6t + (2 * -1 + t2 * -1) + -5t(t + -3) = 0
-8 + 6t + (-2 + -1t2) + -5t(t + -3) = 0

Reorder the terms:
-8 + 6t + -2 + -1t2 + -5t(-3 + t) = 0
-8 + 6t + -2 + -1t2 + (-3 * -5t + t * -5t) = 0
-8 + 6t + -2 + -1t2 + (15t + -5t2) = 0

Reorder the terms:
-8 + -2 + 6t + 15t + -1t2 + -5t2 = 0

Combine like terms: -8 + -2 = -10
-10 + 6t + 15t + -1t2 + -5t2 = 0

Combine like terms: 6t + 15t = 21t
-10 + 21t + -1t2 + -5t2 = 0

Combine like terms: -1t2 + -5t2 = -6t2
-10 + 21t + -6t2 = 0

Solving
-10 + 21t + -6t2 = 0

Solving for variable 't'.

Begin completing the square.  Divide all terms by
-6 the coefficient of the squared term: 

Divide each side by '-6'.
1.666666667 + -3.5t + t2 = 0

Move the constant term to the right:

Add '-1.666666667' to each side of the equation.
1.666666667 + -3.5t + -1.666666667 + t2 = 0 + -1.666666667

Reorder the terms:
1.666666667 + -1.666666667 + -3.5t + t2 = 0 + -1.666666667

Combine like terms: 1.666666667 + -1.666666667 = 0.000000000
0.000000000 + -3.5t + t2 = 0 + -1.666666667
-3.5t + t2 = 0 + -1.666666667

Combine like terms: 0 + -1.666666667 = -1.666666667
-3.5t + t2 = -1.666666667

The t term is -3.5t.  Take half its coefficient (-1.75).
Square it (3.0625) and add it to both sides.

Add '3.0625' to each side of the equation.
-3.5t + 3.0625 + t2 = -1.666666667 + 3.0625

Reorder the terms:
3.0625 + -3.5t + t2 = -1.666666667 + 3.0625

Combine like terms: -1.666666667 + 3.0625 = 1.395833333
3.0625 + -3.5t + t2 = 1.395833333

Factor a perfect square on the left side:
(t + -1.75)(t + -1.75) = 1.395833333

Calculate the square root of the right side: 1.181453906

Break this problem into two subproblems by setting 
(t + -1.75) equal to 1.181453906 and -1.181453906.

Subproblem 1
t + -1.75 = 1.181453906

Simplifying
t + -1.75 = 1.181453906

Reorder the terms:
-1.75 + t = 1.181453906

Solving
-1.75 + t = 1.181453906

Solving for variable 't'.

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

Add '1.75' to each side of the equation.
-1.75 + 1.75 + t = 1.181453906 + 1.75

Combine like terms: -1.75 + 1.75 = 0.00
0.00 + t = 1.181453906 + 1.75
t = 1.181453906 + 1.75

Combine like terms: 1.181453906 + 1.75 = 2.931453906
t = 2.931453906

Simplifying
t = 2.931453906

Subproblem 2
t + -1.75 = -1.181453906

Simplifying
t + -1.75 = -1.181453906

Reorder the terms:
-1.75 + t = -1.181453906

Solving
-1.75 + t = -1.181453906

Solving for variable 't'.

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

Add '1.75' to each side of the equation.
-1.75 + 1.75 + t = -1.181453906 + 1.75

Combine like terms: -1.75 + 1.75 = 0.00
0.00 + t = -1.181453906 + 1.75
t = -1.181453906 + 1.75

Combine like terms: -1.181453906 + 1.75 = 0.568546094
t = 0.568546094

Simplifying
t = 0.568546094

Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {2.931453906, 0.568546094}