Given any two polynomials in a single variable it is always possible to eliminate the variable and obtain a formula showing the relationship between the two polynomials.

Let $p (x) = x^{2} + 2 x$ and $q (x) = x^{2} + x + 1$ then, using a method which does not depend on knowing the answer, show that the relationship between the polynomials is:

$p^{2} - 2 pq + q^{2} + 3 p - 4 q + 3 = 0$