Let's solve the inequality step by step:

The given inequality is:

$-2x^2 + 2x \leq -25$

Step 1: Move all terms to one side.

Add 25 to both sides to bring everything to one side:

$-2x^2 + 2x + 25 \leq 0$

Step 2: Divide through by -2.

To simplify the expression, divide the entire inequality by -2. Remember, dividing by a negative number reverses the inequality sign:

$x^2 - x - \frac{25}{2} \geq 0$

Step 3: Solve the quadratic equation.

At this point, solving the quadratic equation would involve the discriminant:

$x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(-\frac{25}{2})}}{2(1)}$

This expression simplifies to find the roots. However, it looks a bit complicated and likely involves a non-factorable quadratic.

Let me check this more carefullyThe solution to the inequality $x^2 - x - \frac{25}{2} \geq 0$ is:

$x \in (-\infty, -3.07] \cup [4.07, \infty)$

This means that the inequality holds for $x$ values less than or equal to approximately $-3.07$, and for $x$ values greater than or equal to approximately $4.07$.

Do you want more details or have any questions?

Here are 5 related questions for further exploration:

1. How would the solution change if the inequality were strict $>$ instead of $\geq$?
2. What are the exact roots of the quadratic equation?
3. How do we graph the inequality on a number line?
4. What is the role of the discriminant in solving quadratic inequalities?
5. Can we apply this approach to solving cubic inequalities?

Tip: Always remember that dividing an inequality by a negative number reverses the inequality sign!