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To factor the expression \( 3x^2 - 11x - 4 \), let's follow these steps:

### Step 1: Set Up the Equation for Factoring
We want to factor it in the form \((ax + b)(cx + d)\) where:
- The product of the two terms \(a \cdot c\) should equal the coefficient of \(x^2\), which is \(3\).
- The constant term \(b \cdot d\) should equal \(-4\).
- The cross products \(a \cdot d\) and \(b \cdot c\) should combine to give the middle term, \(-11x\).

### Step 2: Multiply the Leading Coefficient and Constant Term
Multiply the leading coefficient (3) by the constant term (-4):
\[
3 \times -4 = -12
\]
We need to find two numbers that multiply to \(-12\) and add up to the middle coefficient, \(-11\).

### Step 3: Find the Pair of Numbers
The numbers that multiply to \(-12\) and add to \(-11\) are \(-12\) and \(+1\).

### Step 4: Rewrite the Middle Term Using These Numbers
Rewrite \(-11x\) as \(-12x + x\):
\[
3x^2 - 12x + x - 4
\]

### Step 5: Factor by Grouping
Group terms and factor each group:
\[
= 3x(x - 4) + 1(x - 4)
\]
Now we can factor out \((x - 4)\):
\[
= (3x + 1)(x - 4)
\]

### Final Answer
The factored form of \(3x^2 - 11x - 4\) is:
\[
(3x + 1)(x - 4)
\]

Would you like further explanation on any of these steps, or do you have any questions?

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Here are 5 related questions to expand on this concept:
1. How would you factor a quadratic expression when the leading coefficient is greater than 1?
2. What are some strategies for finding pairs of numbers that multiply to a specific product and add to a given sum?
3. How can you check if your factorization of a quadratic is correct?
4. What is the significance of factoring in solving quadratic equations?
5. How would you apply factoring techniques to a quadratic in the form \( ax^2 + bx + c = 0 \) for solving it?

**Tip:** When factoring quadratics, always check if you can simplify by factoring out a common factor first.

Learn how to factor the quadratic expression 3x^2 - 11x - 4 with a step-by-step solution. This problem covers factoring techniques, including factoring by grouping, and is ideal for high school students in Grades 9-11.

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