The given expression is $P(-x) = 3(-x)^4 - 7(-x)^2 + 9$. Let's simplify it step by step.

Step-by-Step Simplification:

1. Simplify the powers of $-x$:
   • $(-x)^4 = x^4$, since raising a negative number to an even power results in a positive number.
   • $(-x)^2 = x^2$, for the same reason as above.

So, the expression becomes: $P(-x) = 3x^4 - 7x^2 + 9$

This is the same as the original function $P(x) = 3x^4 - 7x^2 + 9$.

Conclusion:

$P(-x) = P(x)$ This shows that the function is even, meaning it satisfies the property $P(-x) = P(x)$ for all $x$.

Would you like any further details or have any questions about this?

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Here are five related questions you might find interesting:

1. What is the definition of an even function, and how does it differ from an odd function?
2. How can you determine the symmetry of a polynomial function graphically?
3. Can you find the degree and leading coefficient of $P(x) = 3x^4 - 7x^2 + 9$?
4. How would $P(x)$ change if the powers of $x$ were odd instead of even?
5. How would you approach simplifying $P(-x)$ if more complex functions like trigonometric ones were involved?

Tip: Even functions are symmetric about the y-axis, while odd functions exhibit rotational symmetry about the origin.