The given expression in the image is:

$4 - 36u + 81u^2$

To factorize this trinomial, we need to check if it is a perfect square trinomial. A perfect square trinomial is of the form $(ax + b)^2 = a^2x^2 + 2abx + b^2$.

1. Rewriting the expression:

$4 - 36u + 81u^2$

Let's rearrange it in standard quadratic form:

$81u^2 - 36u + 4$

2. Identify $a$, $b$, and $c$:

$a^2 = 81u^2 \Rightarrow a = 9u$ $c^2 = 4 \Rightarrow c = 2$

3. Check the middle term:

$2ab = 2(9u)(2) = 36u$

Since the middle term is $-36u$, we see that the given trinomial can indeed be written as a perfect square trinomial with a negative middle term:

$81u^2 - 36u + 4 = (9u - 2)^2$

Therefore, the factorization of the given perfect square trinomial is:

$(9u - 2)^2$

Would you like further details on the steps or have any other questions?

Relative Questions:

1. What makes a trinomial a perfect square?
2. How do you identify if a trinomial is a perfect square?
3. Can you factorize a perfect square trinomial with a positive middle term?
4. What are the steps to factorize a non-perfect square trinomial?
5. How does the rearrangement of terms help in identifying a perfect square trinomial?
6. Can you provide examples of other perfect square trinomials?
7. How can you verify the factorization of a trinomial?
8. What role does the middle term play in a perfect square trinomial?

Tip:

Always check the middle term to confirm if a trinomial is a perfect square. It should be twice the product of the square roots of the first and last terms.