To determine which polynomial is a perfect square trinomial, we need to identify which one fits the form $(ax + b)^2 = a^2x^2 + 2abx + b^2$.

Let's analyze each option:

1. $9x^2 + 9x + 1$

   • This does not fit the form of $(ax + b)^2$ because $2ab \neq 9x$.

2. $36b^2 - 24b + 8$

   • This does not fit the form of $(ax + b)^2$ because the constant term does not match.

3. $16x^2 + 24x + 9$

   • This seems promising. Let's see if it can be written as $(4x + 3)^2$: $(4x + 3)^2 = (4x)^2 + 2(4x)(3) + 3^2 = 16x^2 + 24x + 9$ This fits the form perfectly.

4. $4a^2 - 10a + 25$

   • This does not fit the form of $(ax + b)^2$ because $2ab \neq -10a$.

Therefore, the polynomial that is a perfect square trinomial is:

$\boxed{16x^2 + 24x + 9}$

Would you like further details or have any questions? Here are some related questions:

1. How do you determine if a polynomial is a perfect square trinomial?
2. Can a perfect square trinomial have negative coefficients?
3. What are some examples of perfect square trinomials?
4. How do you factor a perfect square trinomial?
5. Why is the middle term in a perfect square trinomial twice the product of the square root of the first and last terms?
6. What is the significance of perfect square trinomials in algebra?
7. Can you give an example of a perfect square trinomial with a variable other than $x$?
8. How can you verify if a polynomial is a perfect square trinomial?

Tip: Remember that recognizing patterns like perfect square trinomials can simplify solving quadratic equations significantly.