To determine whether the function $g(x) = x^4 + 3x^2 - 1$ is even, odd, or neither, we can follow these steps:

1. Check for Evenness:

A function $g(x)$ is even if $g(-x) = g(x)$ for all $x$ in the domain.

Let's compute $g(-x)$: $g(-x) = (-x)^4 + 3(-x)^2 - 1$

Simplifying each term: $(-x)^4 = x^4 \quad \text{(since raising to an even power makes it positive)}$ $3(-x)^2 = 3x^2 \quad \text{(again, the even power makes it positive)}$ $-1 \text{ remains the same.}$

Therefore: $g(-x) = x^4 + 3x^2 - 1$

Since $g(-x) = g(x)$, the function $g(x)$ is even.

2. Check for Oddness:

A function $g(x)$ is odd if $g(-x) = -g(x)$ for all $x$ in the domain.

We've already found that $g(-x) = g(x)$, and for the function to be odd, $g(-x)$ should equal $-g(x)$. However: $g(x) = x^4 + 3x^2 - 1$ $-g(x) = -(x^4 + 3x^2 - 1) = -x^4 - 3x^2 + 1$

Clearly, $g(-x) \neq -g(x)$. So, the function is not odd.

Conclusion:

The function ( g