3(p+5)4(p-2)=4(p+6)

Solution for 3(p+5)4(p-2)=4(p+6) equation:

Simplifying
3(p + 5) * 4(p + -2) = 4(p + 6)

Reorder the terms:
3(5 + p) * 4(p + -2) = 4(p + 6)

Reorder the terms:
3(5 + p) * 4(-2 + p) = 4(p + 6)

Reorder the terms for easier multiplication:
3 * 4(5 + p)(-2 + p) = 4(p + 6)

Multiply 3 * 4
12(5 + p)(-2 + p) = 4(p + 6)

Multiply (5 + p) * (-2 + p)
12(5(-2 + p) + p(-2 + p)) = 4(p + 6)
12((-2 * 5 + p * 5) + p(-2 + p)) = 4(p + 6)
12((-10 + 5p) + p(-2 + p)) = 4(p + 6)
12(-10 + 5p + (-2 * p + p * p)) = 4(p + 6)
12(-10 + 5p + (-2p + p2)) = 4(p + 6)

Combine like terms: 5p + -2p = 3p
12(-10 + 3p + p2) = 4(p + 6)
(-10 * 12 + 3p * 12 + p2 * 12) = 4(p + 6)
(-120 + 36p + 12p2) = 4(p + 6)

Reorder the terms:
-120 + 36p + 12p2 = 4(6 + p)
-120 + 36p + 12p2 = (6 * 4 + p * 4)
-120 + 36p + 12p2 = (24 + 4p)

Solving
-120 + 36p + 12p2 = 24 + 4p

Solving for variable 'p'.

Reorder the terms:
-120 + -24 + 36p + -4p + 12p2 = 24 + 4p + -24 + -4p

Combine like terms: -120 + -24 = -144
-144 + 36p + -4p + 12p2 = 24 + 4p + -24 + -4p

Combine like terms: 36p + -4p = 32p
-144 + 32p + 12p2 = 24 + 4p + -24 + -4p

Reorder the terms:
-144 + 32p + 12p2 = 24 + -24 + 4p + -4p

Combine like terms: 24 + -24 = 0
-144 + 32p + 12p2 = 0 + 4p + -4p
-144 + 32p + 12p2 = 4p + -4p

Combine like terms: 4p + -4p = 0
-144 + 32p + 12p2 = 0

Factor out the Greatest Common Factor (GCF), '4'.
4(-36 + 8p + 3p2) = 0

Ignore the factor 4.
Subproblem 1
Set the factor '(-36 + 8p + 3p2)' equal to zero and attempt to solve:

Simplifying
-36 + 8p + 3p2 = 0

Solving
-36 + 8p + 3p2 = 0

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

Divide each side by '3'.
-12 + 2.666666667p + p2 = 0

Move the constant term to the right:

Add '12' to each side of the equation.
-12 + 2.666666667p + 12 + p2 = 0 + 12

Reorder the terms:
-12 + 12 + 2.666666667p + p2 = 0 + 12

Combine like terms: -12 + 12 = 0
0 + 2.666666667p + p2 = 0 + 12
2.666666667p + p2 = 0 + 12

Combine like terms: 0 + 12 = 12
2.666666667p + p2 = 12

The p term is 2.666666667p.  Take half its coefficient (1.333333334).
Square it (1.777777780) and add it to both sides.

Add '1.777777780' to each side of the equation.
2.666666667p + 1.777777780 + p2 = 12 + 1.777777780

Reorder the terms:
1.777777780 + 2.666666667p + p2 = 12 + 1.777777780

Combine like terms: 12 + 1.777777780 = 13.77777778
1.777777780 + 2.666666667p + p2 = 13.77777778

Factor a perfect square on the left side:
(p + 1.333333334)(p + 1.333333334) = 13.77777778

Calculate the square root of the right side: 3.711842909

Break this problem into two subproblems by setting 
(p + 1.333333334) equal to 3.711842909 and -3.711842909.

Subproblem 1
p + 1.333333334 = 3.711842909

Simplifying
p + 1.333333334 = 3.711842909

Reorder the terms:
1.333333334 + p = 3.711842909

Solving
1.333333334 + p = 3.711842909

Solving for variable 'p'.

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

Add '-1.333333334' to each side of the equation.
1.333333334 + -1.333333334 + p = 3.711842909 + -1.333333334

Combine like terms: 1.333333334 + -1.333333334 = 0.000000000
0.000000000 + p = 3.711842909 + -1.333333334
p = 3.711842909 + -1.333333334

Combine like terms: 3.711842909 + -1.333333334 = 2.378509575
p = 2.378509575

Simplifying
p = 2.378509575

Subproblem 2
p + 1.333333334 = -3.711842909

Simplifying
p + 1.333333334 = -3.711842909

Reorder the terms:
1.333333334 + p = -3.711842909

Solving
1.333333334 + p = -3.711842909

Solving for variable 'p'.

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

Add '-1.333333334' to each side of the equation.
1.333333334 + -1.333333334 + p = -3.711842909 + -1.333333334

Combine like terms: 1.333333334 + -1.333333334 = 0.000000000
0.000000000 + p = -3.711842909 + -1.333333334
p = -3.711842909 + -1.333333334

Combine like terms: -3.711842909 + -1.333333334 = -5.045176243
p = -5.045176243

Simplifying
p = -5.045176243

Solution
The solution to the problem is based on the solutions
from the subproblems.
p = {2.378509575, -5.045176243}
Solution
p = {2.378509575, -5.045176243}