To solve this problem, we are asked to factorize the quadratic expression $3x^2 - 6x - 24$.

a) Fill in the gaps

Let's begin by considering the equation in the form: $3x^2 - 6x - 24 = \text{(something)} (x^2 + \text{something} \cdot x + \text{something})$

1. First, we factor out the greatest common factor (GCF) from $3x^2 - 6x - 24$. The GCF is 3, so: $3(x^2 - 2x - 8)$ Now, we need to factor the quadratic $x^2 - 2x - 8$.

2. To factor $x^2 - 2x - 8$, we look for two numbers that multiply to -8 and add up to -2. These numbers are -4 and 2.

Thus, we can write: $x^2 - 2x - 8 = (x - 4)(x + 2)$

3. Therefore, the factorized form of the expression is: $3(x - 4)(x + 2)$

b) Fully factorize $3x^2 - 6x - 24$

Using the result from part (a), the fully factorized form of $3x^2 - 6x - 24$ is: $3(x - 4)(x + 2)$

If you need more details or further steps, feel free to ask!

Further questions:

1. What is the significance of factoring out the greatest common factor?
2. How do you factor a quadratic expression of the form $x^2 + bx + c$?
3. What are the steps involved in solving a quadratic equation using factoring?
4. How can you check the factorization of a quadratic expression by expanding it?
5. What are the different methods of factorization in algebra?

Tip:

Remember to always factor out the GCF first, as this simplifies the problem and makes it easier to factor the remaining terms!