The problem in the image asks to factorise the expression $3x^2 - 12y^2$.

Step-by-Step Solution:

1. Find the common factor:
   The terms $3x^2$ and $-12y^2$ both have a common factor of 3. So, factor out 3: $3(x^2 - 4y^2)$

2. Recognize the difference of squares:
   The expression inside the parentheses $x^2 - 4y^2$ is a difference of squares. This can be written as: $x^2 - (2y)^2$

3. Apply the difference of squares formula:
   The difference of squares formula is: $a^2 - b^2 = (a - b)(a + b)$ Here, $a = x$ and $b = 2y$. So, applying this formula gives: $x^2 - 4y^2 = (x - 2y)(x + 2y)$

4. Final factored form:
   Substitute back into the expression: $3(x^2 - 4y^2) = 3(x - 2y)(x + 2y)$

Final Answer:

$3(x - 2y)(x + 2y)$

Would you like more details on any step, or have any questions?

Here are 5 related questions to expand on this topic:

1. How do you factorise expressions involving cubes instead of squares?
2. Can the difference of squares formula be used for any other types of binomials?
3. What if there were no common factor in the initial expression?
4. How do you factor quadratics that are not perfect squares?
5. How does the factorisation process change with more complex polynomials?

Tip: Always check for common factors before attempting other factoring techniques. This can simplify your work significantly.