How do I find the x-intercepts of an equation like y = x^5 + 9x?

you have to solve for x when y=0, which is not always possible analytically.

There are lots of good graphing apps online, and wolframalpha.com will show the graph and other useful information.

In this case,

y = x(x^4+9)
so y=0 when x=0 or when x^4+9=0
But, x^4 is always positive, so there are no values where x^4 = -9.

So, x=0 is the only solution here.