​I need help with part C. I know a is an even function and b is an odd function. I don’t know what type of function part C is.

C.     h(x)  =   f(x) * g(x) /   x   =   

[ x^4 - 2x^2 ] * [ x^5 + x ] / [2x ]  =

x^2 * [ x^2 -2] * x  * [ x^4 + 1 ]  / [ 2x] =

x^2 * [ x^2 - 2] * [ x^4 + 1 ] / 2  =

[ x^4 - 2x^2 ] [ x^4 + 1 ] / 2 =

[x^8 - 2x^6 + x^4 - 2x^2] /2.....notice that this is an even function because replacing x  with  -x   gives back the same function

Also.....the graph is symmetric with the y axis which is a characteristic of even  functions...here"s the graph :

https://www.desmos.com/calculator/setikvzes3

In general suppose you have an odd function o(x), and an even function e(x).

o(x)o(x) will be even

o(x)e(x) will be odd

e(x)e(x) will be even

here f(x) is even, g(x) is odd, and 1/x is odd

so we have the product of 2 odd and 1 even functions which by above will be even