Make a polynomial identity. using the given equations, demonstrate the proof of polynomial identity through an algebraic proof and a numerical proof in an engaging way.

label and display your new polynomial identity
prove that it is true through an algebraic proof, identifying each step

(x – y)
(x + y)
(y + x)
(y – x)
(x + a)
(y + b)
(x2 + 2xy + y2)
(x2 – 2xy + y2)
(ax + b)
(cy + d)