How to find the x intercepts of a polynomial?

x intercepts are when the function crosses the x-axis, and when the function equals zero. So in this case you would set 3x^4 + 4X^3 + 6x^2 - 4 = 0, and solve for X. The fastest way would be to use a graphing calculator and use the calculate zeros option. (which I did) and your zeros are, x=-0.91 and x=0.64

f(x) on the left side of your equation is the same thing as your Y value.  So yes, setting f(x) (or Y if you want to think about it like that) to 0 will show you where your x intercepts are.  Draw a picture of your Cartesian plane (that's your x-y graph) and draw some random line that weaves back and forth across the X axis as you move from left to right, like for example a sine wave would.  All the spots where your line actually touches the X axis, whether it be in 1 spot or 100 spots, are the X intercepts.  So pick any one of those points and graph it, and your Y value there will be 0 because you haven't moved in the positive or negative direction off of the X axis.  So when you set f(x) to 0, because that's your Y value you're saying that you're not going to move any in the up or down direction.  So you get:
0 = 3x^4 + 4x^3 + 6x^2 - 4.  The shortcut to doing that is using your graphing calculator and graphing that equation.  Everywhere the line crosses the X axis is one of your intercepts