1. **State the problem:** Complete the square for the quadratic equation $$-6x^2 - 582 = -60x$$ and write it in the form $$(x + a)^2 = b$$. 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 the coefficient of $x^2$ to make it 1:** $$\cancel{-6}x^2 + \cancel{-6}(-10x) = \cancel{-6}(-97)$$ which simplifies to $$x^2 - 10x = -97$$ 4. **Complete the square:** Take half the coefficient of $x$, square it, and add to both sides. Half of $-10$ is $-5$, and $(-5)^2 = 25$. Add 25 to both sides: $$x^2 - 10x + 25 = -97 + 25$$ 5. **Simplify the right side:** $$x^2 - 10x + 25 = -72$$ 6. **Write the left side as a perfect square:** $$(x - 5)^2 = -72$$ **Final intermediate step:** $$(x - 5)^2 = -72$$