2t(4-t)+3(5-t)=2(t+1)+2t(t-6)+(3t-1)(1-t)-3t^2

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Solution for 2t(4-t)+3(5-t)=2(t+1)+2t(t-6)+(3t-1)(1-t)-3t^2 equation: 


Simplifying
2t(4 + -1t) + 3(5 + -1t) = 2(t + 1) + 2t(t + -6) + (3t + -1)(1 + -1t) + -3t2
(4 * 2t + -1t * 2t) + 3(5 + -1t) = 2(t + 1) + 2t(t + -6) + (3t + -1)(1 + -1t) + -3t2
(8t + -2t2) + 3(5 + -1t) = 2(t + 1) + 2t(t + -6) + (3t + -1)(1 + -1t) + -3t2
8t + -2t2 + (5 * 3 + -1t * 3) = 2(t + 1) + 2t(t + -6) + (3t + -1)(1 + -1t) + -3t2
8t + -2t2 + (15 + -3t) = 2(t + 1) + 2t(t + -6) + (3t + -1)(1 + -1t) + -3t2

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

Combine like terms: 8t + -3t = 5t
15 + 5t + -2t2 = 2(t + 1) + 2t(t + -6) + (3t + -1)(1 + -1t) + -3t2

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

Reorder the terms:
15 + 5t + -2t2 = 2 + 2t + 2t(-6 + t) + (3t + -1)(1 + -1t) + -3t2
15 + 5t + -2t2 = 2 + 2t + (-6 * 2t + t * 2t) + (3t + -1)(1 + -1t) + -3t2
15 + 5t + -2t2 = 2 + 2t + (-12t + 2t2) + (3t + -1)(1 + -1t) + -3t2

Reorder the terms:
15 + 5t + -2t2 = 2 + 2t + -12t + 2t2 + (-1 + 3t)(1 + -1t) + -3t2

Multiply (-1 + 3t) * (1 + -1t)
15 + 5t + -2t2 = 2 + 2t + -12t + 2t2 + (-1(1 + -1t) + 3t * (1 + -1t)) + -3t2
15 + 5t + -2t2 = 2 + 2t + -12t + 2t2 + ((1 * -1 + -1t * -1) + 3t * (1 + -1t)) + -3t2
15 + 5t + -2t2 = 2 + 2t + -12t + 2t2 + ((-1 + 1t) + 3t * (1 + -1t)) + -3t2
15 + 5t + -2t2 = 2 + 2t + -12t + 2t2 + (-1 + 1t + (1 * 3t + -1t * 3t)) + -3t2
15 + 5t + -2t2 = 2 + 2t + -12t + 2t2 + (-1 + 1t + (3t + -3t2)) + -3t2

Combine like terms: 1t + 3t = 4t
15 + 5t + -2t2 = 2 + 2t + -12t + 2t2 + (-1 + 4t + -3t2) + -3t2

Reorder the terms:
15 + 5t + -2t2 = 2 + -1 + 2t + -12t + 4t + 2t2 + -3t2 + -3t2

Combine like terms: 2 + -1 = 1
15 + 5t + -2t2 = 1 + 2t + -12t + 4t + 2t2 + -3t2 + -3t2

Combine like terms: 2t + -12t = -10t
15 + 5t + -2t2 = 1 + -10t + 4t + 2t2 + -3t2 + -3t2

Combine like terms: -10t + 4t = -6t
15 + 5t + -2t2 = 1 + -6t + 2t2 + -3t2 + -3t2

Combine like terms: 2t2 + -3t2 = -1t2
15 + 5t + -2t2 = 1 + -6t + -1t2 + -3t2

Combine like terms: -1t2 + -3t2 = -4t2
15 + 5t + -2t2 = 1 + -6t + -4t2

Solving
15 + 5t + -2t2 = 1 + -6t + -4t2

Solving for variable 't'.

Reorder the terms:
15 + -1 + 5t + 6t + -2t2 + 4t2 = 1 + -6t + -4t2 + -1 + 6t + 4t2

Combine like terms: 15 + -1 = 14
14 + 5t + 6t + -2t2 + 4t2 = 1 + -6t + -4t2 + -1 + 6t + 4t2

Combine like terms: 5t + 6t = 11t
14 + 11t + -2t2 + 4t2 = 1 + -6t + -4t2 + -1 + 6t + 4t2

Combine like terms: -2t2 + 4t2 = 2t2
14 + 11t + 2t2 = 1 + -6t + -4t2 + -1 + 6t + 4t2

Reorder the terms:
14 + 11t + 2t2 = 1 + -1 + -6t + 6t + -4t2 + 4t2

Combine like terms: 1 + -1 = 0
14 + 11t + 2t2 = 0 + -6t + 6t + -4t2 + 4t2
14 + 11t + 2t2 = -6t + 6t + -4t2 + 4t2

Combine like terms: -6t + 6t = 0
14 + 11t + 2t2 = 0 + -4t2 + 4t2
14 + 11t + 2t2 = -4t2 + 4t2

Combine like terms: -4t2 + 4t2 = 0
14 + 11t + 2t2 = 0

Factor a trinomial.
(7 + 2t)(2 + t) = 0


Subproblem 1
Set the factor '(7 + 2t)' equal to zero and attempt to solve:

Simplifying
7 + 2t = 0

Solving
7 + 2t = 0

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

Add '-7' to each side of the equation.
7 + -7 + 2t = 0 + -7

Combine like terms: 7 + -7 = 0
0 + 2t = 0 + -7
2t = 0 + -7

Combine like terms: 0 + -7 = -7
2t = -7

Divide each side by '2'.
t = -3.5

Simplifying
t = -3.5

Subproblem 2
Set the factor '(2 + t)' equal to zero and attempt to solve:

Simplifying
2 + t = 0

Solving
2 + t = 0

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

Add '-2' to each side of the equation.
2 + -2 + t = 0 + -2

Combine like terms: 2 + -2 = 0
0 + t = 0 + -2
t = 0 + -2

Combine like terms: 0 + -2 = -2
t = -2

Simplifying
t = -2
Solution
t = {-3.5, -2}

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