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Let's solve each inequality analytically.

### Given inequalities:
1. **(a)** \( x^2 + 4x + 3 \geq 0 \)
2. **(b)** \( x^2 + 4x + 3 < 0 \)

### **Step-by-Step Solution**

#### 1. **Factor the quadratic expression: \( x^2 + 4x + 3 \)**
   The expression can be factored as follows:
   \[
   x^2 + 4x + 3 = (x + 3)(x + 1)
   \]
   Therefore, the factored form is \( (x + 3)(x + 1) \).

#### 2. **Find the roots of the quadratic equation**
   Setting \( (x + 3)(x + 1) = 0 \):
   \[
   x + 3 = 0 \quad \text{or} \quad x + 1 = 0
   \]
   Thus, the roots are:
   \[
   x = -3 \quad \text{and} \quad x = -1
   \]

### **Solution for (a): \( x^2 + 4x + 3 \geq 0 \)**
   The inequality is \( (x + 3)(x + 1) \geq 0 \).

   - Identify the intervals based on the roots:
     - \( (-\infty, -3) \)
     - \( (-3, -1) \)
     - \( (-1, \infty) \)

   - Choose a test point in each interval to determine the sign:
     - For \( (-\infty, -3) \), choose \( x = -4 \): \( (-4+3)(-4+1) = (-)(-) = + \)
     - For \( (-3, -1) \), choose \( x = -2 \): \( (-2+3)(-2+1) = (+)(-) = - \)
     - For \( (-1, \infty) \), choose \( x = 0 \): \( (0+3)(0+1) = (+)(+) = + \)

   - Thus, the sign of the expression in each interval is:
     - \( (-\infty, -3) \): Positive
     - \( (-3, -1) \): Negative
     - \( (-1, \infty) \): Positive

   - Solution: \( x \leq -3 \) or \( x \geq -1 \).
   - In interval notation: \( (-\infty, -3] \cup [-1, \infty) \).

### **Solution for (b): \( x^2 + 4x + 3 < 0 \)**
   The inequality is \( (x + 3)(x + 1) < 0 \).

   - From the interval analysis above, the expression is negative only in the interval:
     - \( (-3, -1) \).

   - Solution: \( -3 < x < -1 \).
   - In interval notation: \( (-3, -1) \).

### **Final Answers**
1. **(a)** \( (-\infty, -3] \cup [-1, \infty) \)
2. **(b)** \( (-3, -1) \)

Would you like further details or have any questions?

### 5 Related Questions
1. How do you determine the intervals to test the sign of a quadratic expression?
2. What is the significance of roots in solving inequalities?
3. How would the solution change if the inequality sign was reversed for part (a)?
4. Can you explain the graphical interpretation of the inequality solutions?
5. How would you solve inequalities with non-factorable quadratic expressions?

### Tip:
Always test points in each interval determined by the roots to ensure the correct sign when solving inequalities.

Solve the inequalities (a) x^2 + 4x + 3 ≥ 0 and (b) x^2 + 4x + 3 < 0.

Learn how to solve the quadratic inequalities x^2 + 4x + 3 ≥ 0 and x^2 + 4x + 3 < 0 with a detailed step-by-step solution. This example uses factoring and interval testing to determine the correct solutions for both inequalities. Suitable for students in Grades 9-11.

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