Let's solve each part of the factorization step-by-step:

(a) Factorize $x^2 - 4$

This is a difference of squares: $x^2 - 4 = (x - 2)(x + 2)$

(b) Factorize $3x^2 - 11x - 4$

For this quadratic, we will use factoring by splitting the middle term. We need to find two numbers that multiply to $3 \times (-4) = -12$ and add up to $-11$. The numbers are $-12$ and $1$.

Now, split the middle term: $3x^2 - 12x + x - 4$ Factor by grouping: $3x(x - 4) + 1(x - 4)$ Factor out the common binomial: $(3x + 1)(x - 4)$

(c) Factorize $3x^2 + 4x - 4$

For this quadratic, we again factor by splitting the middle term. We need to find two numbers that multiply to $3 \times (-4) = -12$ and add to $4$. The numbers are $6$ and $-2$.

Now, split the middle term: $3x^2 + 6x - 2x - 4$ Factor by grouping: $3x(x + 2) - 2(x + 2)$ Factor out the common binomial: $(3x - 2)(x + 2)$

Final answers:

• (a) $(x - 2)(x + 2)$
• (b) $(3x + 1)(x - 4)$
• (c) $(3x - 2)(x + 2)$

These factorizations match the options in the image.

Would you like more details, or do you have any questions?

Here are five related questions to deepen your understanding:

1. What is the difference of squares, and when can it be applied?
2. How do you determine which method to use when factoring quadratics?
3. Can every quadratic expression be factored, or are there exceptions?
4. How does the discriminant of a quadratic equation affect its factorization?
5. Why is grouping used when factoring quadratics with a leading coefficient greater than 1?

Tip: Always check your factored expression by expanding it back to ensure correctness.