	The function is given as f (x) = x2 + 4x + 3. Determine the interval where the graph of f (x) is decreasing.
 (–∞, –2)
 (–∞, –3)
 (–2, ∞)
 (–3, ∞)

The function given is \( f(x) = x^2 + 4x + 3 \). To determine the interval where the function is decreasing, we need to analyze the behavior of its derivative.

1. **Find the derivative of \( f(x) \):**
   \[
   f'(x) = \frac{d}{dx}(x^2 + 4x + 3) = 2x + 4
   \]

2. **Find the critical point where the derivative is zero:**
   \[
   2x + 4 = 0
   \]
   Solving for \( x \):
   \[
   x = -2
   \]

   This critical point, \( x = -2 \), is where the function changes its behavior (increasing or decreasing).

3. **Analyze the sign of the derivative:**
   - For \( x < -2 \), the derivative \( f'(x) = 2x + 4 \) is negative, which means the function is decreasing.
   - For \( x > -2 \), the derivative \( f'(x) = 2x + 4 \) is positive, which means the function is increasing.

Thus, the function is decreasing on the interval \( (-\infty, -2) \).

### Correct answer: \( (-\infty, -2) \)

Would you like further details on this, or have any questions?

Here are 5 related questions to expand your understanding:
1. How do you find intervals where a function is increasing?
2. What does it mean when the derivative of a function is zero?
3. How can the second derivative help determine the concavity of a function?
4. What are the critical points of a function, and how are they found?
5. How do you graph a quadratic function using its vertex and intercepts?

**Tip:** Always check the derivative's sign to determine where a function is increasing or decreasing.

The function is given as f(x) = x^2 + 4x + 3. Determine the interval where the graph of f(x) is decreasing.
 (-∞, -2)
 (-∞, -3)
 (-2, ∞)
 (-3, ∞)

Learn how to find the interval where the quadratic function f(x) = x^2 + 4x + 3 is decreasing. This step-by-step solution involves calculus concepts such as derivatives and the First Derivative Test. Suitable for students in Grades 10-12.

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