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.
