1. **State the problem:** We want to rewrite the quadratic equation $$-6x^2 - 582 = -60x$$ in the form $$(x + a)^2 = b$$ by completing the square. 2. **Rewrite the equation:** Move all terms to one side and isolate the quadratic and linear terms: $$-6x^2 + 60x = 582$$ 3. **Divide both sides by -6** to make the coefficient of $x^2$ equal to 1: $$\cancel{-6}x^2 + \cancel{-6}(-10x) = \cancel{-6}(-97)$$ which simplifies to $$x^2 - 10x = -97$$ 4. **Complete the square:** Take half of the coefficient of $x$, which is $-10$, divide by 2 to get $-5$, then square it to get $25$. 5. Add $25$ to both sides to complete the square: $$x^2 - 10x + 25 = -97 + 25$$ 6. Simplify the right side: $$x^2 - 10x + 25 = -72$$ 7. Write the left side as a perfect square: $$(x - 5)^2 = -72$$ **Final intermediate step:** $$(x - 5)^2 = -72$$