To factor the given perfect square trinomial $4g^2 - 4g + 1$, we need to identify it as a square of a binomial.

1. Write down the trinomial: $4g^2 - 4g + 1$

2. Recognize the structure of a perfect square trinomial $a^2 - 2ab + b^2$, which factors to $(a - b)^2$.

3. Identify $a^2$ and $b^2$: $a^2 = 4g^2 \quad \Rightarrow \quad a = 2g$ $b^2 = 1 \quad \Rightarrow \quad b = 1$

4. Confirm that the middle term fits the form $-2ab$: $-2ab = -2 \cdot 2g \cdot 1 = -4g$

Since the middle term $-4g$ matches, we can conclude that the trinomial is indeed a perfect square trinomial. Therefore, it factors to: $(2g - 1)^2$

So, the factored form of $4g^2 - 4g + 1$ is: $(2g - 1)^2$

Would you like further details or have any questions?

Here are some related questions:

1. What are the characteristics of perfect square trinomials?
2. How can you distinguish a perfect square trinomial from other trinomials?
3. Can a perfect square trinomial have a negative leading coefficient?
4. How do you derive the general formula for factoring perfect square trinomials?
5. What steps can be taken to factor trinomials that do not fit the perfect square pattern?
6. How do you apply the completing the square method to factor trinomials?
7. What are some common mistakes to avoid when factoring trinomials?
8. How do you verify if your factored form of a trinomial is correct?

Tip: When factoring perfect square trinomials, always verify each term to ensure it fits the $a^2 - 2ab + b^2$ or $a^2 + 2ab + b^2$ pattern.