MathNet 2003 — Part 02 Question 30

Let $a, b, c$ be real numbers such that $a^{2} + b^{2} + c^{2} = 1$. Prove that $P = ab + bc + ca - 2(a + b + c) \geq -\frac{5}{2}$. Are there values of $a, b, c$ such that $P = -\frac{5}{2}$?