To factor the quadratic expression \(3x^2 + 19x + 6\), we will look for two numbers that multiply to give the product of the coefficient of \(x^2\) (which is 3) and the constant term (which is 6), and also add up to the coefficient of \(x\) (which is 19).

### Step-by-Step Solution:

1. **Find the Product and Sum:**

   The product of the coefficient of \(x^2\) (3) and the constant term (6) is:
   \[
   3 \times 6 = 18
   \]

   We need two numbers that multiply to 18 and add up to 19. These numbers are 18 and 1.

2. **Rewrite the Middle Term:**

   Rewrite the middle term (19x) using 18 and 1:
   \[
   3x^2 + 18x + x + 6
   \]

3. **Group the Terms:**

   Group the terms to factor by grouping:
   \[
   (3x^2 + 18x) + (x + 6)
   \]

   Factor out the common factors from each group:
   \[
   3x(x + 6) + 1(x + 6)
   \]

4. **Factor Out the Common Binomial:**

   Notice that \((x + 6)\) is a common factor:
   \[
   (3x + 1)(x + 6)
   \]

### Final Answer:

The factored form of the expression \(3x^2 + 19x + 6\) is:
\[
\boxed{(3x + 1)(x + 6)}
\]

Would you like more details or have any questions? Here are some related topics:

1. How to factor other quadratic expressions?
2. What is the quadratic formula and how to use it?
3. How to solve quadratic equations by completing the square?
4. Why is factoring useful in solving quadratic equations?
5. How to recognize special factoring patterns, like perfect square trinomials?

**Tip:** Always check your factorization by expanding the factors back out to ensure they match the original expression.

Discover how to factor the quadratic expression 3x^2 + 19x + 6 with this comprehensive step-by-step solution. Learn key algebraic concepts, including factorization by grouping, and see how to break down the middle term. Perfect for high school students in Grades 9-11.

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