Find the remainder when the polynomial function $f(x) = x^3 - 7x + 12$ is divided by $x + 3$ using the Remainder Theorem. Rewrite the divisor $x + 3$ in the form $x - c$ to find that $x - (-3)$, so $c = -3$. By the Remainder Theorem, the remainder is $f(c)$. Evaluate the function at $c = -3$: $f(-3) = (-3)^3 - 7(-3) + 12$. Simplifying this gives $-27 + 21 + 12$, which equals $6$. The remainder is $6$.