I saw to prove that $a^2+b^2=c^2$ you can produce the algebraic solutions $a = x^2-y^2, b = 2xy, c = x^2+y^2$. To check that works just multiply it out but it produces infinitely many solutions when you set x and y to be integers.
Can you please show me more examples of this, especially if they show how to find the polynomials!