Solve x^2+2xy+y^2-p^2=2[(x+y+p)+b]

$$x ^ { 2 } + 2 x y + y ^ { 2 } - p ^ { 2 } = 2 [ ( x + y + p ) + b ]$$

Solve for b

$$b=-\frac{\left(2+p-y-x\right)\left(x+y+p\right)}{2}$$

Steps for Solving Linear Equation

Use the distributive property to multiply 2 by x+y+p+b.

$$x^{2}+2xy+y^{2}-p^{2}=2x+2y+2p+2b$$

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

$$2x+2y+2p+2b=x^{2}+2xy+y^{2}-p^{2}$$

Subtract 2x from both sides.

$$2y+2p+2b=x^{2}+2xy+y^{2}-p^{2}-2x$$

Subtract 2y from both sides.

$$2p+2b=x^{2}+2xy+y^{2}-p^{2}-2x-2y$$

Subtract 2p from both sides.

$$2b=x^{2}+2xy+y^{2}-p^{2}-2x-2y-2p$$

The equation is in standard form.

$$2b=x^{2}+2xy-2x+y^{2}-2y-p^{2}-2p$$

Divide both sides by 2.

$$\frac{2b}{2}=\frac{\left(x+y-p-2\right)\left(x+y+p\right)}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$b=\frac{\left(x+y-p-2\right)\left(x+y+p\right)}{2}$$

Solve for p (complex solution)

$$p=-\sqrt{x^{2}+2xy-2x+y^{2}-2y-2b+1}-1$$

$$p=\sqrt{x^{2}+2xy-2x+y^{2}-2y-2b+1}-1$$

Solve for p

$$p=-\sqrt{x^{2}+2xy-2x+y^{2}-2y-2b+1}-1$$

$$p=\sqrt{x^{2}+2xy-2x+y^{2}-2y-2b+1}-1\text{, }b\leq xy+\frac{\left(x-1\right)^{2}}{2}+\frac{y^{2}}{2}-y$$