One way of getting a less complicated answer is by replacing 'x^2' with 's', and 'y^2' with 't', that is:

s = x^2

t = y^2

And then solving the following system:

Click here to see the step by step solving

So, now you have the solutions for 's' and 't':

s = 4

t = 4

Now, we need to go back to 'x' and 'y':

x^2 = 4

y^2 = 4

We have the following solutions for 'x':

x1 = 2

x2 = -2

And the following solutions for y"

y1 = 2

y2 = -2

So, there are 4 solutions: (2, 2), (-2, 2), (-2, 2) and (-2, -2).

You can also see the solutions if you graph the equations - there are 4 interceptions, thus 4 solutions:

Click here to see the graph.