Age 16 to 18 Challenge Level:
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 $p(x) = x^2 + 2x$ 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:
\[ p^2 - 2pq + q^2 + 3p - 4q + 3 = 0 \]