Here are the solutions for each of the inequalities you've provided:

1. $x^2 - 9x + 8 \geq 0$ Factorize: $(x - 1)(x - 8) \geq 0$ Critical points: $x = 1$ and $x = 8$

   The solution is: $x \leq 1$ or $x \geq 8$.

2. $x^2 - 35x + 300 \leq 0$ Factorize: $(x - 15)(x - 20) \leq 0$ Critical points: $x = 15$ and $x = 20$

   The solution is: $15 \leq x \leq 20$.

3. $-2x^2 + 7x + 4 \leq 0$ Rewrite as: $2x^2 - 7x - 4 \geq 0$ Factorize: $(2x + 1)(x - 4) \geq 0$ Critical points: $x = -\frac{1}{2}$ and $x = 4$

   The solution is: $x \leq -\frac{1}{2}$ or $x \geq 4$.

4. $2x^2 + 7x - 10 \leq x^2 + 12$ Simplify: $x^2 + 7x - 22 \leq 0$ Factorize: $(x - 2)(x + 11) \leq 0$ Critical points: $x = 2$ and $x = -11$

   The solution is: $-11 \leq x \leq 2$.

5. $-4x^2 - x + 2 \geq 0$ Rewrite: $4x^2 + x - 2 \leq 0$ Factorize: $(4x - 3)(x + 2) \leq 0$ Critical points: $x = \frac{3}{4}$ and $x = -2$

   The solution is: $-2 \leq x \leq \frac{3}{4}$.

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Would you like a detailed breakdown of any of these steps? Here are some follow-up questions for further practice:

1. How do we determine the critical points in these inequalities?
2. Why do we change the sign of inequalities when multiplying or dividing by a negative?
3. What are the rules for solving quadratic inequalities?
4. How can graphing help visualize the solutions to inequalities?
5. How would you solve an inequality involving absolute values?

Tip: Always check the intervals created by critical points to determine the correct signs of the inequality in each region.