Based on the degree, what is the greatest number of zeros the polynomial f(x)=5x^4+3x^2 could have?

To determine the greatest number of zeros a polynomial can have, we can utilize the Fundamental Theorem of Algebra, which states that a polynomial of degree \( n \) has exactly \( n \) roots (including both real and complex roots, and counting multiplicities).

For the polynomial given:

\[
f(x) = 5x^4 + 3x^2
\]

we first need to identify its degree. The highest exponent of \( x \) in this polynomial is \( 4 \) (from the term \( 5x^4 \)). Thus, the degree of this polynomial is \( 4 \).

Since the degree of the polynomial is \( 4 \), it can have at most \( 4 \) zeros (roots). These zeros can be a combination of real and complex numbers.

Next, we can consider the possibility of finding the actual zeros of the polynomial. To do this, we can factor \( f(x) \) as follows:

1. Factor out the common term \( x^2 \):

\[
f(x) = 5x^4 + 3x^2 = x^2(5x^2 + 3)
\]

2. Now analyze \( x^2 \) and \( 5x^2 + 3 \):

- The factor \( x^2 = 0 \) gives a double root at \( x = 0 \).
- For \( 5x^2 + 3 = 0 \):

\[
5x^2 + 3 = 0 \Rightarrow 5x^2 = -3 \Rightarrow x^2 = -\frac{3}{5}
\]

This indicates that \( x = \pm i \sqrt{\frac{3}{5}} \) are complex roots.

Thus, we have:

- A double root at \( x = 0 \) (counting as \( 2 \) roots)
- Two complex roots \( x = i \sqrt{\frac{3}{5}} \) and \( x = -i \sqrt{\frac{3}{5}} \)

In total, we have \( 4 \) roots (including multiplicities): two from \( x^2 = 0 \) and two from \( 5x^2 + 3 = 0 \).

In conclusion, regardless of whether the zeros are real or complex, the greatest number of zeros the polynomial \( f(x) = 5x^4 + 3x^2 \) can have is:

\[
\boxed{4}
\]