1. **State the problem:** We want to find the maximum value of the objective function $$z = -200x^2 + 4y^2$$ subject to the constraint $$y = 5x - 10$$. 2. **Use substitution:** Since $$y = 5x - 10$$, substitute this into the objective function to express $$z$$ in terms of $$x$$ only: $$z = -200x^2 + 4(5x - 10)^2$$. 3. **Expand and simplify:** $$z = -200x^2 + 4(25x^2 - 100x + 100)$$ $$z = -200x^2 + 100x^2 - 400x + 400$$ $$z = (-200x^2 + 100x^2) - 400x + 400$$ $$z = -100x^2 - 400x + 400$$. 4. **Find critical points:** To find the maximum, take the derivative of $$z$$ with respect to $$x$$ and set it to zero: $$\frac{dz}{dx} = -200x - 400 = 0$$ 5. **Solve for $$x$$:** $$-200x = 400$$ $$x = -2$$. 6. **Find corresponding $$y$$:** $$y = 5(-2) - 10 = -10 - 10 = -20$$. 7. **Calculate maximum value of $$z$$:** $$z = -200(-2)^2 + 4(-20)^2 = -200(4) + 4(400) = -800 + 1600 = 800$$. **Final answer:** The maximum value of $$z$$ is $$800$$.