What expression squared gives $4x^2$? That will be your $a$.

After choosing $a$, use the middle term $-2ab$ to solve for $b$, then verify that $b^2$ matches the constant term.

In Desmos, enter $f(x)=4x^2-12x+9$.

Enter each option as $g_1(x)=(2x-3)^2$, $g_2(x)=(2x+3)^2$, $g_3(x)=(x-3)^2$, and $g_4(x)=4(x-3)^2$.

The equivalent expression will produce a graph that lies exactly on top of $f(x)$ for all $x$. The matching option is $(2x - 3)^{2}$.

Compare $4x^{2}-12x+9$ to the identity

Here, $4x^2$ suggests $a=2x$ because $(2x)^2=4x^2$.

Using $a=2x$, the middle term of $(a-b)^2$ is $-2ab=-2(2x)b=-4bx$.

Set $-4bx$ equal to $-12x$ to find $b$:

Check the constant term: $b^2=3^2=9$, which matches.

Since $a=2x$ and $b=3$, the expression factors as

So the equivalent expression is $(2x - 3)^{2}$.