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.