p(p+q)-q(p+q)=0

Solution for p(p+q)-q(p+q)=0 equation:

Simplifying
p(p + q) + -1q(p + q) = 0
(p * p + q * p) + -1q(p + q) = 0

Reorder the terms:
(pq + p2) + -1q(p + q) = 0
(pq + p2) + -1q(p + q) = 0
pq + p2 + (p * -1q + q * -1q) = 0
pq + p2 + (-1pq + -1q2) = 0

Reorder the terms:
pq + -1pq + p2 + -1q2 = 0

Combine like terms: pq + -1pq = 0
0 + p2 + -1q2 = 0
p2 + -1q2 = 0

Solving
p2 + -1q2 = 0

Solving for variable 'p'.

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

Add 'q2' to each side of the equation.
p2 + -1q2 + q2 = 0 + q2

Combine like terms: -1q2 + q2 = 0
p2 + 0 = 0 + q2
p2 = 0 + q2
Remove the zero:
p2 = q2

Simplifying
p2 = q2

Take the square root of each side:
p = {-1q, q}