1. **State the problem:** We want to rewrite the equation $$-3x^2 + 49x = -5x + 183$$ in the form $$(x + a)^2 = b$$ by completing the square. 2. **Rearrange the equation:** Move all terms to one side and group the $x^2$ and $x$ terms together: $$-3x^2 + 49x + 5x = 183$$ $$-3x^2 + 54x = 183$$ 3. **Factor out the coefficient of $x^2$ from the left side:** $$-3(x^2 - 18x) = 183$$ 4. **Complete the square inside the parentheses:** Take half of the coefficient of $x$, which is $-18$, half is $-9$, then square it: $$(-9)^2 = 81$$ Add and subtract 81 inside the parentheses: $$-3(x^2 - 18x + 81 - 81) = 183$$ 5. **Rewrite as a perfect square and simplify:** $$-3((x - 9)^2 - 81) = 183$$ Distribute $-3$: $$-3(x - 9)^2 + 243 = 183$$ 6. **Isolate the perfect square term:** $$-3(x - 9)^2 = 183 - 243$$ $$-3(x - 9)^2 = -60$$ Divide both sides by $-3$: $$\cancel{-3}(x - 9)^2 = \cancel{-3} \times 20$$ $$(x - 9)^2 = 20$$ **Final intermediate step:** $$(x - 9)^2 = 20$$