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The given inequalities are:

96. (a) \(x^2 + 6x + 8 < 0\)

(b) \(x^2 + 6x + 8 \geq 0\)

### Let's solve them one by one.

#### Part (a): \(x^2 + 6x + 8 < 0\)

1. **Factor the quadratic expression:**

   \[
   x^2 + 6x + 8 = (x + 4)(x + 2)
   \]

2. The inequality becomes:

   \[
   (x + 4)(x + 2) < 0
   \]

3. **Solve the inequality using sign analysis:**

   To determine when this product is less than zero, we find the critical points by setting each factor equal to zero:

   \[
   x + 4 = 0 \quad \Rightarrow \quad x = -4
   \]
   \[
   x + 2 = 0 \quad \Rightarrow \quad x = -2
   \]

   These are the points where the expression changes sign. Now, analyze the sign of \((x + 4)(x + 2)\) in the intervals formed by these points: \((- \infty, -4)\), \((-4, -2)\), and \((-2, \infty)\).

   - For \(x \in (-\infty, -4)\): Both factors are negative, so their product is positive.
   - For \(x \in (-4, -2)\): One factor is negative and the other is positive, so the product is negative.
   - For \(x \in (-2, \infty)\): Both factors are positive, so their product is positive.

   The expression \((x + 4)(x + 2) < 0\) holds in the interval where the product is negative, which is:

   \[
   (-4, -2)
   \]

   Therefore, the solution to part (a) is:

   \[
   x \in (-4, -2)
   \]

#### Part (b): \(x^2 + 6x + 8 \geq 0\)

1. Using the same factorization, the inequality becomes:

   \[
   (x + 4)(x + 2) \geq 0
   \]

2. The critical points are \(x = -4\) and \(x = -2\).

3. Analyze the intervals formed by these points, again checking where the product is greater than or equal to zero:
   
   - For \(x \in (-\infty, -4)\): The product is positive.
   - For \(x = -4\): The product is zero.
   - For \(x \in (-4, -2)\): The product is negative.
   - For \(x = -2\): The product is zero.
   - For \(x \in (-2, \infty)\): The product is positive.

   Therefore, the solution to \( (x + 4)(x + 2) \geq 0 \) is:

   \[
   x \in (-\infty, -4] \cup [-2, \infty)
   \]

### Final answers:
- (a) \(x \in (-4, -2)\)
- (b) \(x \in (-\infty, -4] \cup [-2, \infty)\)

Would you like more details or clarifications on any step?

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Here are 5 related questions:
1. How do you solve a quadratic inequality if the quadratic cannot be factored?
2. Can a quadratic inequality have no solution?
3. How would the inequality change if it were \(x^2 + 6x + 8 > 0\)?
4. How can we solve inequalities graphically?
5. What are the implications of a non-strict inequality, like \(\geq\) or \(\leq\), in terms of solutions?

**Tip:** Always check the sign of each factor in the intervals between critical points when solving inequalities by factoring.

Solve the inequalities: (a) x^2 + 6x + 8 < 0 and (b) x^2 + 6x + 8 ≥ 0

Learn how to solve quadratic inequalities such as x^2 + 6x + 8 < 0 and x^2 + 6x + 8 ≥ 0. This step-by-step solution involves factoring, sign analysis, and identifying the intervals where the inequalities hold true. Perfect for students in Grades 9-10.

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