To find the possible integer roots of the polynomial $f(x) = x^5 + 9x^2 - 4x + 4$, we can apply the Rational Root Theorem.

The Rational Root Theorem states that any rational solution $\frac{p}{q}$ must satisfy:

• $p$ is a divisor of the constant term (the last term).
• $q$ is a divisor of the leading coefficient (the coefficient of the highest power of $x$).

For $f(x) = x^5 + 9x^2 - 4x + 4$:

• The constant term is $4$, and its divisors are: $\pm 1, \pm 2, \pm 4$.
• The leading coefficient is $1$, and its divisors are: $\pm 1$.

Therefore, the possible integer roots (since $q = \pm 1$) are the divisors of $4$:

$\boxed{1, -1, 2, -2, 4, -4}$

These are the candidate integer roots.

Do you want more details on how to test these roots or any follow-up questions? Here are 5 related questions:

1. How can we use synthetic division to test these candidate roots?
2. What is the Rational Root Theorem, and why does it work?
3. How would you test non-integer roots for this equation?
4. What happens if none of these candidate roots work?
5. What methods can be used to find non-integer roots of a polynomial?

Tip: Synthetic division is a quick way to test if a candidate root is valid for a polynomial equation.