f(x) is a notation for a "function of x." Usually it is written as f(x)= "something", which is an expression with x as what we call the independent variable. Many times f(x) and y are the same thing.
In solving a quadratic equation you are looking for the roots of the expression in the form of a^2 + bx + c. And the roots are defined as the values where this expression equal zero (on a graph, it's where the curve crosses the x-axis).
So what you are looking for is the values of x that satisfy x^2 + 2x - 3 = 0. (I'm guessing your teacher means f(x) = x^2 + 2x - 3)
To start focus on the portion of this expression x^2 + 2x. Now ask yourself if you can find a number that when added to this will allow you to write it as a quantity squared i.e (x + k)^2. So we want (x + k)^2 which is the same as x^2 + 2xk + k^2 = x^2 + 2x. It looks like k = 1.
So if we go back to the original equation, x^2 + 2x - 3 = 0. If we add k^2, that is 1, to both sides we get x^2 + 2x + 1 - 3 = 1. Now make the simplification you just determined: (x + 1)^2 - 3 = 1. (You just completed the square). Rearranging: (x - 1)^2 = 4.
Can you solve this from here?
Be sure to check your answers (there are two of them) by substituting them, one at a time, back into the original equation.
The next topic you are likely to study is the quadratic equation. It is based on this method.