1. **State the problem:** We want to rewrite the quadratic equation $$-3x^2 + 6x = -321$$ in the form $$(x + a)^2 = b$$ by completing the square. 2. **Rewrite the equation:** First, divide both sides by $-3$ to make the coefficient of $x^2$ equal to 1. $$-3x^2 + 6x = -321$$ $$\Rightarrow \cancel{-3}x^2 + \cancel{-3} \cdot (-2x) = \cancel{-3} \cdot 107$$ $$x^2 - 2x = 107$$ 3. **Complete the square:** Take half of the coefficient of $x$, which is $-2$, half is $-1$, then square it: $$(-1)^2 = 1$$. Add and subtract 1 on the left side to complete the square: $$x^2 - 2x + 1 - 1 = 107$$ Group the perfect square trinomial: $$(x - 1)^2 - 1 = 107$$ 4. **Isolate the perfect square:** Add 1 to both sides: $$(x - 1)^2 = 108$$ **Final intermediate step:** $$(x - 1)^2 = 108$$