2p(3p+1)=12

Solution for 2p(3p+1)=12 equation:

Simplifying
2p(3p + 1) = 12

Reorder the terms:
2p(1 + 3p) = 12
(1 * 2p + 3p * 2p) = 12
(2p + 6p2) = 12

Solving
2p + 6p2 = 12

Solving for variable 'p'.

Reorder the terms:
-12 + 2p + 6p2 = 12 + -12

Combine like terms: 12 + -12 = 0
-12 + 2p + 6p2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-6 + p + 3p2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-6 + p + 3p2)' equal to zero and attempt to solve:

Simplifying
-6 + p + 3p2 = 0

Solving
-6 + p + 3p2 = 0

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

Divide each side by '3'.
-2 + 0.3333333333p + p2 = 0

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + 0.3333333333p + 2 + p2 = 0 + 2

Reorder the terms:
-2 + 2 + 0.3333333333p + p2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + 0.3333333333p + p2 = 0 + 2
0.3333333333p + p2 = 0 + 2

Combine like terms: 0 + 2 = 2
0.3333333333p + p2 = 2

The p term is p.  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.
0.3333333333p + 0.25 + p2 = 2 + 0.25

Reorder the terms:
0.25 + 0.3333333333p + p2 = 2 + 0.25

Combine like terms: 2 + 0.25 = 2.25
0.25 + 0.3333333333p + p2 = 2.25

Factor a perfect square on the left side:
(p + 0.5)(p + 0.5) = 2.25

Calculate the square root of the right side: 1.5

Break this problem into two subproblems by setting 
(p + 0.5) equal to 1.5 and -1.5.

Subproblem 1
p + 0.5 = 1.5

Simplifying
p + 0.5 = 1.5

Reorder the terms:
0.5 + p = 1.5

Solving
0.5 + p = 1.5

Solving for variable 'p'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + p = 1.5 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + p = 1.5 + -0.5
p = 1.5 + -0.5

Combine like terms: 1.5 + -0.5 = 1
p = 1

Simplifying
p = 1

Subproblem 2
p + 0.5 = -1.5

Simplifying
p + 0.5 = -1.5

Reorder the terms:
0.5 + p = -1.5

Solving
0.5 + p = -1.5

Solving for variable 'p'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + p = -1.5 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + p = -1.5 + -0.5
p = -1.5 + -0.5

Combine like terms: -1.5 + -0.5 = -2
p = -2

Simplifying
p = -2

Solution
The solution to the problem is based on the solutions
from the subproblems.
p = {1, -2}
Solution
p = {1, -2}