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The problem requires factoring the quadratic expression \(4x^2 + 8x + 3\) using the **grouping method**, also known as the **AC method**. Here's the solution step-by-step:

### Step 1: Multiply \(a\) and \(c\) (from \(ax^2 + bx + c\))
Here, \(a = 4\), \(b = 8\), and \(c = 3\).
\[
a \cdot c = 4 \cdot 3 = 12
\]

### Step 2: Find two numbers that multiply to \(a \cdot c = 12\) and add to \(b = 8\)
The two numbers are **6** and **2** because:
\[
6 \cdot 2 = 12 \quad \text{and} \quad 6 + 2 = 8
\]

### Step 3: Rewrite the middle term (\(8x\)) using these two numbers
\[
4x^2 + 6x + 2x + 3
\]

### Step 4: Group terms in pairs
\[
(4x^2 + 6x) + (2x + 3)
\]

### Step 5: Factor out the greatest common factor (GCF) from each group
\[
2x(2x + 3) + 1(2x + 3)
\]

### Step 6: Factor out the common binomial factor
\[
(2x + 3)(2x + 1)
\]

### Final Factorization
The factorization of \(4x^2 + 8x + 3\) is:
\[
(2x + 3)(2x + 1)
\]

Let me know if you'd like more explanation or a walkthrough of other examples!  

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### Related Questions:
1. How is the AC method applied for negative coefficients?
2. What if no two numbers satisfy the product-sum relationship? What happens then?
3. How does factoring by grouping compare to other factoring techniques?
4. Can this quadratic expression be solved for its roots after factoring?
5. How do we verify if our factorization is correct?

### Tip:
Always double-check your factorization by expanding the factors to ensure you get back to the original expression.

Factor the quadratic expression 4x^2 + 8x + 3 using the grouping method (AC method).

Step-by-step guide to factoring the quadratic expression 4x^2 + 8x + 3 using the grouping method, also known as the AC method. Includes detailed breakdowns and explanations for algebra students in Grades 8-10.

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