Solve 4p^2-1/p^2-1=pq+q/4p-2

$$\frac { 4 p ^ { 2 } - 1 } { p ^ { 2 } - 1 } = \frac { p q + q } { 4 p - 2 }$$

Solve for q

$$q=-\frac{2\times \left(\frac{2p-1}{p+1}\right)^{2}\left(2p+1\right)}{1-p}$$

$$p\neq \frac{1}{2}\text{ and }|p|\neq 1$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 2\left(p-1\right)\left(2p-1\right)\left(p+1\right), the least common multiple of p^{2}-1,4p-2.

$$\left(4p-2\right)\left(4p^{2}-1\right)=\left(p^{2}-1\right)\left(pq+q\right)$$

Use the distributive property to multiply 4p-2 by 4p^{2}-1.

$$16p^{3}-4p-8p^{2}+2=\left(p^{2}-1\right)\left(pq+q\right)$$

Use the distributive property to multiply p^{2}-1 by pq+q.

$$16p^{3}-4p-8p^{2}+2=qp^{3}+p^{2}q-pq-q$$

Swap sides so that all variable terms are on the left hand side.

$$qp^{3}+p^{2}q-pq-q=16p^{3}-4p-8p^{2}+2$$

Combine all terms containing q.

$$\left(p^{3}+p^{2}-p-1\right)q=16p^{3}-4p-8p^{2}+2$$

The equation is in standard form.

$$\left(p^{3}+p^{2}-p-1\right)q=16p^{3}-8p^{2}-4p+2$$

Divide both sides by p^{3}+p^{2}-p-1.

$$\frac{\left(p^{3}+p^{2}-p-1\right)q}{p^{3}+p^{2}-p-1}=\frac{2\left(2p+1\right)\left(2p-1\right)^{2}}{p^{3}+p^{2}-p-1}$$

Dividing by p^{3}+p^{2}-p-1 undoes the multiplication by p^{3}+p^{2}-p-1.

$$q=\frac{2\left(2p+1\right)\left(2p-1\right)^{2}}{p^{3}+p^{2}-p-1}$$

Divide 2\left(1+2p\right)\left(-1+2p\right)^{2} by p^{3}+p^{2}-p-1.

$$q=\frac{2\left(2p+1\right)\left(2p-1\right)^{2}}{\left(p-1\right)\left(p+1\right)^{2}}$$