To factor the expression $3x^2 - 11x - 4$, we can follow these steps:

1. Identify the terms: The expression is in the form $ax^2 + bx + c$ where:

   • $a = 3$
   • $b = -11$
   • $c = -4$

2. Multiply $a$ and $c$:

   • $3 \times -4 = -12$

3. Find factors of $-12$ that add up to $b = -11$:

   • The factors of $-12$ that add to $-11$ are $-12$ and $1$ (since $-12 + 1 = -11$).

4. Rewrite the middle term: Use $-12$ and $1$ to split the middle term, so: $3x^2 - 12x + x - 4$

5. Factor by grouping:

   • Group terms: $(3x^2 - 12x) + (x - 4)$
   • Factor out the common factors from each group: $3x(x - 4) + 1(x - 4)$
   • Now, factor out $(x - 4)$: $(3x + 1)(x - 4)$

6. Final answer: The factored form is: $(3x + 1)(x - 4)$

Would you like further clarification on any step?

Here are some related questions for practice:

1. How would you factor $4x^2 - 15x + 9$?
2. What if the middle term was positive, such as in $3x^2 + 11x - 4$?
3. How can you confirm if your factorization is correct?
4. How would you approach factoring if $a = 1$ in $x^2 - 11x - 4$?
5. What is the difference between factoring by grouping and other factoring methods?

Tip: Always double-check your factors by expanding them to see if you return to the original expression.