Is my work transforming $9(x+2)^2$ to $(3x+6)^2$ correct? What is this method called? $$9(x+2)^2 = 3^2(x+2)^2=(3(x+2))^2=(3x+6)^2$$
I want to know if the use of brackets in this problem has been done correctly. What is this method called?

Yes, all steps are correct.
You can also verify that the result is correct by expanding both expressions.
The first expression expands out to
$$9(x+2)^2=9(x^2+2\cdot 2\cdot x+2^2)=9(x^2+4x+4)=9x^2+36x+36$$
while the final expression expands out to
$$(3x+6)^2=(3x)^2+2\cdot(3x)\cdot 6 + 6^2 = 3^2x^2+2\cdot3\cdot6\cdot x + 36=9x^2+36x+36$$
The two expansions match, confirming the original expressions are equivalent.
