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The quadratic expression given is:

\[
3x^2 + 12x + 4
\]

### To graph this quadratic function, we can follow these steps:

1. **Identify the coefficients**: 
   - \(a = 3\), \(b = 12\), and \(c = 4\).

2. **Find the vertex**:
   The vertex form of a quadratic is \(y = a(x-h)^2 + k\), where \((h, k)\) is the vertex. We can find the vertex using:
   
   \[
   h = \frac{-b}{2a} = \frac{-12}{2(3)} = -2
   \]
   
   Plug \(x = -2\) into the equation to find \(k\):
   
   \[
   k = 3(-2)^2 + 12(-2) + 4 = 3(4) - 24 + 4 = 12 - 24 + 4 = -8
   \]
   
   So, the vertex is \((-2, -8)\).

3. **Determine the axis of symmetry**:
   The axis of symmetry is the vertical line through the vertex, given by \(x = -2\).

4. **Find the y-intercept**:
   The y-intercept occurs when \(x = 0\):
   
   \[
   y = 3(0)^2 + 12(0) + 4 = 4
   \]
   
   So, the y-intercept is \((0, 4)\).

5. **Plot additional points**:
   You can calculate other points by substituting different values of \(x\) and finding the corresponding \(y\) values.

### Would you like me to graph this equation, or do you need further details on how the quadratic behaves?

Here are 5 related questions for further exploration:

1. How do you find the roots of a quadratic equation?
2. What is the discriminant of this quadratic and what does it tell us?
3. How can you complete the square for this quadratic equation?
4. How would the graph change if the coefficient of \(x^2\) was negative?
5. How do we find the focus and directrix of a parabola from its equation?

**Tip**: The sign of the coefficient of \(x^2\) determines whether the parabola opens upwards or downwards.

Learn how to solve and graph the quadratic equation 3x^2 + 12x + 4. This problem includes finding the vertex, axis of symmetry, y-intercept, and key points for graphing, as well as a step-by-step explanation for students in Grades 8-10.

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