Determine the number of x-intercepts that appear on a graph of each function.f (x) = (x - 6)2(x + 2)2

Short answer There are 2 places where the graph touches the x axis just as you said.
I take this to mean f(x) = (x - 6)^2*(x + 2)^2

 
There are two places where the graph touches the x axis but that is not an intercept. 
When a graph just touches the x axis, that counts as a root. As the graph shows, these 2 places are
x = 6 and
x = - 2 is where they touch.

If you have a graphing calculator, you can see what this looks like for yourself. The window settings are 
x min = - 3
x max = 7
y min = 0
y max = 250. 

There is a local maximum between x = -2 and x = 6.
So the answer, once again is at x = -2 and x = 6 and that makes 2 roots for the equation.