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

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


Simplifying
(2p + 3q)(p + -4q) + (p + 6q)(p + 2q) = 0

Multiply (2p + 3q) * (p + -4q)
(2p * (p + -4q) + 3q * (p + -4q)) + (p + 6q)(p + 2q) = 0
((p * 2p + -4q * 2p) + 3q * (p + -4q)) + (p + 6q)(p + 2q) = 0

Reorder the terms:
((-8pq + 2p2) + 3q * (p + -4q)) + (p + 6q)(p + 2q) = 0
((-8pq + 2p2) + 3q * (p + -4q)) + (p + 6q)(p + 2q) = 0
(-8pq + 2p2 + (p * 3q + -4q * 3q)) + (p + 6q)(p + 2q) = 0
(-8pq + 2p2 + (3pq + -12q2)) + (p + 6q)(p + 2q) = 0

Reorder the terms:
(-8pq + 3pq + 2p2 + -12q2) + (p + 6q)(p + 2q) = 0

Combine like terms: -8pq + 3pq = -5pq
(-5pq + 2p2 + -12q2) + (p + 6q)(p + 2q) = 0

Multiply (p + 6q) * (p + 2q)
-5pq + 2p2 + -12q2 + (p(p + 2q) + 6q * (p + 2q)) = 0
-5pq + 2p2 + -12q2 + ((p * p + 2q * p) + 6q * (p + 2q)) = 0

Reorder the terms:
-5pq + 2p2 + -12q2 + ((2pq + p2) + 6q * (p + 2q)) = 0
-5pq + 2p2 + -12q2 + ((2pq + p2) + 6q * (p + 2q)) = 0
-5pq + 2p2 + -12q2 + (2pq + p2 + (p * 6q + 2q * 6q)) = 0
-5pq + 2p2 + -12q2 + (2pq + p2 + (6pq + 12q2)) = 0

Reorder the terms:
-5pq + 2p2 + -12q2 + (2pq + 6pq + p2 + 12q2) = 0

Combine like terms: 2pq + 6pq = 8pq
-5pq + 2p2 + -12q2 + (8pq + p2 + 12q2) = 0

Reorder the terms:
-5pq + 8pq + 2p2 + p2 + -12q2 + 12q2 = 0

Combine like terms: -5pq + 8pq = 3pq
3pq + 2p2 + p2 + -12q2 + 12q2 = 0

Combine like terms: 2p2 + p2 = 3p2
3pq + 3p2 + -12q2 + 12q2 = 0

Combine like terms: -12q2 + 12q2 = 0
3pq + 3p2 + 0 = 0
3pq + 3p2 = 0

Solving
3pq + 3p2 = 0

Solving for variable 'p'.

Factor out the Greatest Common Factor (GCF), '3p'.
3p(q + p) = 0

Ignore the factor 3.

Subproblem 1
Set the factor 'p' equal to zero and attempt to solve:

Simplifying
p = 0

Solving
p = 0

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

Simplifying
p = 0

Subproblem 2
Set the factor '(q + p)' equal to zero and attempt to solve:

Simplifying
q + p = 0

Reorder the terms:
p + q = 0

Solving
p + q = 0

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

Add '-1q' to each side of the equation.
p + q + -1q = 0 + -1q

Combine like terms: q + -1q = 0
p + 0 = 0 + -1q
p = 0 + -1q
Remove the zero:
p = -1q

Simplifying
p = -1q
Solution
p = {0, -1q}

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