(2p+1)(2q+1)=2(2pq+p+q)+1

Solution for (2p+1)(2q+1)=2(2pq+p+q)+1 equation:

Simplifying
(2p + 1)(2q + 1) = 2(2pq + p + q) + 1

Reorder the terms:
(1 + 2p)(2q + 1) = 2(2pq + p + q) + 1

Reorder the terms:
(1 + 2p)(1 + 2q) = 2(2pq + p + q) + 1

Multiply (1 + 2p) * (1 + 2q)
(1(1 + 2q) + 2p * (1 + 2q)) = 2(2pq + p + q) + 1
((1 * 1 + 2q * 1) + 2p * (1 + 2q)) = 2(2pq + p + q) + 1
((1 + 2q) + 2p * (1 + 2q)) = 2(2pq + p + q) + 1
(1 + 2q + (1 * 2p + 2q * 2p)) = 2(2pq + p + q) + 1
(1 + 2q + (2p + 4pq)) = 2(2pq + p + q) + 1

Reorder the terms:
(1 + 2p + 4pq + 2q) = 2(2pq + p + q) + 1
(1 + 2p + 4pq + 2q) = 2(2pq + p + q) + 1

Reorder the terms:
1 + 2p + 4pq + 2q = 2(p + 2pq + q) + 1
1 + 2p + 4pq + 2q = (p * 2 + 2pq * 2 + q * 2) + 1
1 + 2p + 4pq + 2q = (2p + 4pq + 2q) + 1

Reorder the terms:
1 + 2p + 4pq + 2q = 1 + 2p + 4pq + 2q

Add '-1' to each side of the equation.
1 + 2p + 4pq + -1 + 2q = 1 + 2p + 4pq + -1 + 2q

Reorder the terms:
1 + -1 + 2p + 4pq + 2q = 1 + 2p + 4pq + -1 + 2q

Combine like terms: 1 + -1 = 0
0 + 2p + 4pq + 2q = 1 + 2p + 4pq + -1 + 2q
2p + 4pq + 2q = 1 + 2p + 4pq + -1 + 2q

Reorder the terms:
2p + 4pq + 2q = 1 + -1 + 2p + 4pq + 2q

Combine like terms: 1 + -1 = 0
2p + 4pq + 2q = 0 + 2p + 4pq + 2q
2p + 4pq + 2q = 2p + 4pq + 2q

Add '-2p' to each side of the equation.
2p + 4pq + -2p + 2q = 2p + 4pq + -2p + 2q

Reorder the terms:
2p + -2p + 4pq + 2q = 2p + 4pq + -2p + 2q

Combine like terms: 2p + -2p = 0
0 + 4pq + 2q = 2p + 4pq + -2p + 2q
4pq + 2q = 2p + 4pq + -2p + 2q

Reorder the terms:
4pq + 2q = 2p + -2p + 4pq + 2q

Combine like terms: 2p + -2p = 0
4pq + 2q = 0 + 4pq + 2q
4pq + 2q = 4pq + 2q

Add '-4pq' to each side of the equation.
4pq + -4pq + 2q = 4pq + -4pq + 2q

Combine like terms: 4pq + -4pq = 0
0 + 2q = 4pq + -4pq + 2q
2q = 4pq + -4pq + 2q

Combine like terms: 4pq + -4pq = 0
2q = 0 + 2q
2q = 2q

Add '-2q' to each side of the equation.
2q + -2q = 2q + -2q

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

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

Solving
0 = 0

Couldn't find a variable to solve for.

This equation is an identity, all real numbers are solutions.