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?

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.