Exploring Quadratic Expressions and Inequalities in Variables

a) Derive the formula for $(a + b + c)^2$.
b) Given that $a + b + c = 5$ and $ab + ac + bc = 5$, determine the possible values of $a^2 + b^2 + c^2$.
c) If $x + y + z = 0$, prove that $xy + yz + zx \leq 0$.