Let's solve each inequality step by step:

(f) $-x^2 - 5x - 6 \geq 0$

1. Rewrite it in standard quadratic form: $x^2 + 5x + 6 \leq 0$

2. Factorize the quadratic expression: $x^2 + 5x + 6 = (x + 2)(x + 3)$

3. Solve the equality $(x + 2)(x + 3) = 0$, which gives: $x = -2 \quad \text{and} \quad x = -3$

4. Use these roots to divide the real number line into intervals: $(-\infty, -3), \ (-3, -2), \ (-2, \infty)$

5. Test the sign of $(x + 2)(x + 3)$ in each interval:

   • For $x \in (-\infty, -3)$, $(x + 2)(x + 3) > 0$.
   • For $x \in (-3, -2)$, $(x + 2)(x + 3) < 0$.
   • For $x \in (-2, \infty)$, $(x + 2)(x + 3) > 0$.

6. Since $x^2 + 5x + 6 \leq 0$, the solution is: $x \in [-3, -2]$

(i) $x(x - 1) \leq 20$

1. Rewrite as: $x^2 - x - 20 \leq 0$

2. Factorize the quadratic expression: $x^2 - x - 20 = (x - 5)(x + 4)$

3. Solve the equality $(x - 5)(x + 4) = 0$, which gives: $x = 5 \quad \text{and} \quad x = -4$

4. Use these roots to divide the real number line into intervals: $(-\infty, -4), \ (-4, 5), \ (5, \infty)$

5. Test the sign of $(x - 5)(x + 4)$ in each interval:

   • For $x \in (-\infty, -4)$, $(x - 5)(x + 4) > 0$.
   • For $x \in (-4, 5)$, $(x - 5)(x + 4) < 0$.
   • For $x \in (5, \infty)$, $(x - 5)(x + 4) > 0$.

6. Since $x^2 - x - 20 \leq 0$, the solution is: $x \in [-4, 5]$

(l) $x^2 + 3x \geq -2$

1. Rewrite as: $x^2 + 3x + 2 \geq 0$

2. Factorize the quadratic expression: $x^2 + 3x + 2 = (x + 1)(x + 2)$

3. Solve the equality $(x + 1)(x + 2) = 0$, which gives: $x = -1 \quad \text{and} \quad x = -2$

4. Use these roots to divide the real number line into intervals: $(-\infty, -2), \ (-2, -1), \ (-1, \infty)$

5. Test the sign of $(x + 1)(x + 2)$ in each interval:

   • For $x \in (-\infty, -2)$, $(x + 1)(x + 2) > 0$.
   • For $x \in (-2, -1)$, $(x + 1)(x + 2) < 0$.
   • For $x \in (-1, \infty)$, $(x + 1)(x + 2) > 0$.

6. Since $x^2 + 3x + 2 \geq 0$, the solution is: $x \in (-\infty, -2] \cup [-1, \infty)$

Final Solutions:

• (f): $x \in [-3, -2]$
• (i): $x \in [-4, 5]$
• (l): $x \in (-\infty, -2] \cup [-1, \infty)$

Let me know if you'd like further details!

Related Questions:

1. What is the method to graph solutions for quadratic inequalities?
2. How do test points help in determining the intervals of a quadratic inequality?
3. What does the sign of a quadratic expression depend on?
4. Why is the factorization method essential in solving quadratic inequalities?
5. How would the solution change for strict inequalities (e.g., $<$ or $>$)?

Tip:

Always double-check the direction of the inequality after testing intervals to ensure correct results.