To factor the trinomial \( 4x^2 + 8x + 3 \), follow these steps:

### Step 1: Identify the coefficients
- \( a = 4 \) (coefficient of \( x^2 \))
- \( b = 8 \) (coefficient of \( x \))
- \( c = 3 \) (constant term)

### Step 2: Find two numbers that multiply to \( a \times c = 4 \times 3 = 12 \) and add to \( b = 8 \)
We need to find two numbers that multiply to 12 and add to 8. The numbers 6 and 2 work because:
- \( 6 \times 2 = 12 \)
- \( 6 + 2 = 8 \)

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

### Step 4: Factor by grouping
Group the terms into two pairs and factor out the greatest common factor (GCF) from each group:
\[
(4x^2 + 6x) + (2x + 3)
\]
Factor out the GCF from each group:
\[
2x(2x + 3) + 1(2x + 3)
\]

### Step 5: Factor out the common binomial factor
Now, factor out the common binomial factor \( 2x + 3 \):
\[
(2x + 3)(2x + 1)
\]

### Final Answer
The factored form of the trinomial \( 4x^2 + 8x + 3 \) is:
\[
(2x + 3)(2x + 1)
\]

Would you like more details or have any questions? 

Here are some related questions:
1. How would you factor a trinomial where the leading coefficient is 1?
2. Can you explain the process of factoring by grouping?
3. What happens if you cannot find factors that add to the middle coefficient?
4. How does the quadratic formula relate to factoring trinomials?
5. How can you verify that your factorization is correct?

**Tip:** Always check your factored form by expanding it back out to ensure it equals the original trinomial.

Learn how to factor the trinomial 4x^2 + 8x + 3 with a detailed step-by-step guide. This problem covers key algebraic concepts, including factoring quadratic trinomials. Suitable for students in Grades 7-9.

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