To factor the quadratic expression $4x^2 - 13x - 12$, we'll use the method of splitting the middle term.

Steps:

1. Multiply the coefficient of $x^2$ (which is 4) by the constant term (which is -12): $4 \times -12 = -48$

2. Find two numbers that multiply to -48 and add up to the coefficient of $x$ (which is -13): $-16 \text{ and } 3$ (Because $-16 \times 3 = -48$ and $-16 + 3 = -13$)

3. Rewrite the middle term $-13x$ using $-16x$ and $3x$: $4x^2 - 16x + 3x - 12$

4. Group the terms in pairs: $(4x^2 - 16x) + (3x - 12)$

5. Factor out the greatest common factor (GCF) from each pair: $4x(x - 4) + 3(x - 4)$

6. Factor out the common binomial factor $(x - 4)$: $(4x + 3)(x - 4)$

So, the factored form of $4x^2 - 13x - 12$ is: $(4x + 3)(x - 4)$

Would you like further details or have any questions?

Here are some related questions you might find interesting:

1. How do you factor a trinomial when the leading coefficient is not 1?
2. Can all quadratic expressions be factored using this method?
3. What is the difference between factoring and expanding an expression?
4. How do you factor a quadratic expression with complex coefficients?
5. What is the quadratic formula, and how is it related to factoring?
6. How would you check if your factored expression is correct?
7. Can you factor a quadratic expression that does not have rational roots?
8. What is the significance of the discriminant in factoring quadratics?

Tip: Always double-check your factorization by expanding the factors to see if you get the original expression back.