Final answer:
To factor the polynomial completely, we look for common factors. After factoring out the common factor, the quadratic expression inside the parentheses is prime.
Step-by-step explanation:
To factor the polynomial 25a²b² + 30ab + 9 completely, we can look for common factors among the terms. Notice that all the terms have a common factor of 3. Factoring out the common factor of 3, we get:
3(8a²b² + 10ab + 3)
Now, we can try to factor the quadratic expression inside the parentheses. However, after trying various factorizations, we find that the quadratic expression is prime and cannot be factored further. Therefore, the completely factored form of the polynomial is:
3(8a²b² + 10ab + 3)