$$P(x)$$ is a polynomial in $$x$$ with real coefficients. Given that the polynomial $$P^2(x) + (9x - 2007)^2$$ has a real root $$\alpha$$, determine $$\alpha$$ and also the multiplicity of $$\alpha$$.

PLACEHOLDER ANSWER - the question has two answers. A sum of two real squares is zero only if both are zero, so $$9\alpha - 2007 = 0$$ and $$P(\alpha) = 0$$, giving $$\alpha = 223$$. Write $$P(x) = (x - 223)^k Q(x)$$ with $$k \ge 1$$ and $$Q(223) \ne 0$$; then $$P^2(x) + 81(x - 223)^2 = (x - 223)^2\left[(x - 223)^{2k - 2}Q^2(x) + 81\right]$$. The bracket is positive at $$x = 223$$, so the multiplicity of $$\alpha$$ is exactly 2.