Factorise completely 3x3-12xy2

The terms are $3x^3$ and $-12xy^2$.
The numerical coefficients are 3 and -12. The GCF of 3 and 12 is 3.
The variable parts are $x^3$ and $x$. The GCF of $x^3$ and $x$ is $x$.
There is no common factor for $y^2$ as it only appears in the second term.
So, the GCF of the entire expression is $3x$.

Divide each term by $3x$:
$\frac{3x^3}{3x} = x^2$
$\frac{-12xy^2}{3x} = -4y^2$
So, the expression becomes $3x(x^2 - 4y^2)$.

The term inside the parentheses, $x^2 - 4y^2$, is in the form of $a^2 - b^2$.
Here, $a = x$ and $b = \sqrt{4y^2} = 2y$.
The difference of squares identity is $a^2 - b^2 = (a-b)(a+b)$.
So, $x^2 - 4y^2 = (x - 2y)(x + 2y)$.

Substitute the factored form of $x^2 - 4y^2$ back into the expression from Step 2.

The completely factorised expression is $3x(x - 2y)(x + 2y)$.

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