according to the rational root theorem, $-\frac{2}{5}$ is a potential rational root of which…

Explanation:

Step1: Recall Rational Root Theorem

The Rational Root Theorem states that if a polynomial $f(x)=a_nx^n + a_{n - 1}x^{n-1}+\cdots+a_1x + a_0$ has integer coefficients, then a potential rational root is of the form $\frac{p}{q}$, where $p$ is a factor of the constant term $a_0$ and $q$ is a factor of the leading - coefficient $a_n$. For the root $-\frac{2}{5}$, $p=\pm2$ and $q = \pm5$. So the leading - coefficient $a_n$ should be a multiple of 5 and the constant term $a_0$ should be a multiple of 2.

Step2: Check each function

For $f(x)=4x^4 - 7x^2+x + 25$:

The leading - coefficient $a_n = 4$ is not a multiple of 5, so this is not the correct function.

For $f(x)=9x^4 - 7x^2+x + 10$:

The leading - coefficient $a_n = 9$ is not a multiple of 5, so this is not the correct function.

For $f(x)=10x^4 - 7x^2+x + 9$:

The leading - coefficient $a_n = 10$ is a multiple of 5 and the constant term $a_0 = 9$ is not a multiple of 2, so this is not the correct function.

For $f(x)=25x^4 - 7x^2+x + 4$:

The leading - coefficient $a_n = 25$ is a multiple of 5 and the constant term $a_0 = 4$ is a multiple of 2.

Answer:

$f(x)=25x^4 - 7x^2+x + 4$