The problem asks us to factor the trinomial $4x^2 - 8x + 4$ in the form of $(Ax - B)^2$, where $A$ and $B$ are constants to be determined.

Let's start by recognizing that this is a perfect square trinomial. We want to express $4x^2 - 8x + 4$ as the square of a binomial. Here’s how we approach this:

1. Identify the coefficients:

   • The first term is $4x^2$, which is the square of $2x$, so $A = 2$.
   • The last term is $4$, which is the square of $2$, so $B = 2$.

2. Check the middle term: The middle term should be twice the product of $A$ and $B$, i.e., $-2 \times A \times B$. $-2 \times 2 \times 2 = -8x$ This matches the middle term in the original trinomial.

Therefore, we can factor the trinomial as: $4x^2 - 8x + 4 = (2x - 2)^2$

Thus, $A = 2$ and $B = 2$.

Final Answer:

• $A = 2$
• $B = 2$

Would you like more details on how perfect square trinomials are factored? Here are some related questions:

1. How do you identify a perfect square trinomial?
2. Can this technique be applied to any trinomial?
3. What happens if the middle term doesn't match twice the product of $A$ and $B$?
4. How do you check if a trinomial is factored correctly?
5. Can we factor this trinomial in other ways besides a perfect square?

Tip: Always verify the middle term when factoring perfect square trinomials to avoid errors.