Given that n^2+2pn+p^2+q^2=r^2 has real roots show that r^2=q^2

It doesn't seem true, and here's a counterexample: observe that the first three terms form a perfect square. You can rewrite the equation as

This is basically the Pythagorean theorem applied to a triangle with sides n+p, q and r. For example, pick:

The expression becomes

Which is true, even if

The constants, -3 and -8, are like terms

The terms 3p and p are like terms

The terms in the expression are p^2, -3, 3p, -8, p, p^3

The expression contains 6 terms.

Like terms have the same variables raised to the same powers.