To find the range of the quadratic function \( f(x) = 4x^2 + 4x + 3 \), we can first rewrite it in vertex form by completing the square.

1. **Identify the coefficients**:
   - \( a = 4 \)
   - \( b = 4 \)
   - \( c = 3 \)

2. **Complete the square**:
   \[
   f(x) = 4(x^2 + x) + 3
   \]
   Take half of the coefficient of \( x \) (which is \( 1 \)), square it (\( \left(\frac{1}{2}\right)^2 = \frac{1}{4} \)), and add and subtract this inside the parentheses:
   \[
   f(x) = 4\left(x^2 + x + \frac{1}{4} - \frac{1}{4}\right) + 3
   \]
   Simplifying gives:
   \[
   f(x) = 4\left(\left(x + \frac{1}{2}\right)^2 - \frac{1}{4}\right) + 3
   \]
   \[
   f(x) = 4\left(x + \frac{1}{2}\right)^2 - 1 + 3
   \]
   \[
   f(x) = 4\left(x + \frac{1}{2}\right)^2 + 2
   \]

3. **Determine the vertex**:
   The vertex form \( f(x) = 4\left(x + \frac{1}{2}\right)^2 + 2 \) shows that the vertex is at \( \left(-\frac{1}{2}, 2\right) \).

4. **Analyze the parabola**:
   Since \( a = 4 > 0 \), the parabola opens upwards. The minimum value of \( f(x) \) occurs at the vertex.

5. **Find the range**:
   The minimum value is \( 2 \), and since the parabola opens upwards, the range is:
   \[
   [2, \infty)
   \]

### Conclusion
The range of the function \( f(x) = 4x^2 + 4x + 3 \) is \([2, \infty)\).

Do you want any details or have any questions? Here are some related questions you might consider:

1. How would you find the vertex of a different quadratic function?
2. What is the significance of the discriminant in a quadratic equation?
3. How do you determine if a quadratic function opens upward or downward?
4. Can you explain how to graph a quadratic function?
5. What are some applications of quadratic functions in real life?

**Tip**: When completing the square, always remember to balance the equation by adding and subtracting the same value to maintain equality.

Explore how to determine the range of the quadratic function f(x) = 4x^2 + 4x + 3 using vertex form and completing the square. This guide is suitable for students in Grades 8-10 and covers essential algebraic concepts.

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