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To factor the quadratic expression \(3x^2 + 14x + 8\), we need to find two binomials \((x + m)(3x + n)\) that multiply to give the original quadratic expression.

### Step 1: Multiply the coefficient of \(x^2\) term (3) and the constant term (8).
\[
3 \times 8 = 24
\]

### Step 2: Find two numbers that multiply to 24 and add up to 14.
The numbers are 12 and 2, because:
\[
12 \times 2 = 24 \quad \text{and} \quad 12 + 2 = 14
\]

### Step 3: Rewrite the middle term \(14x\) using these numbers.
\[
3x^2 + 12x + 2x + 8
\]

### Step 4: Factor by grouping.
\[
3x(x + 4) + 2(x + 4)
\]

### Step 5: Factor out the common factor \((x + 4)\).
\[
(x + 4)(3x + 2)
\]

Thus, the factored form is:
\[
(3x + 2)(x + 4)
\]

Would you like further details, or do you have any questions?

Here are 5 related questions to expand your understanding:
1. How do you factor a quadratic equation if the coefficient of \(x^2\) is 1?
2. What are the different methods for factoring quadratic equations?
3. How can you verify that a quadratic equation has been factored correctly?
4. What is the significance of the discriminant in factoring quadratic equations?
5. How can completing the square be used to factor quadratics?

**Tip:** When factoring quadratics, always check if there is a greatest common factor (GCF) before proceeding with other methods.

Factor the quadratic expression 3x^2 + 14x + 8.

Learn how to factor the quadratic expression 3x^2 + 14x + 8 with a detailed step-by-step explanation. This problem includes factoring by grouping and understanding the relationship between coefficients and the factored form. Suitable for students in Grades 8-10.

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