Problem #16: Mexican Table Mats

My son gave me a set of table mats he bought in Mexico. They were rectangular and were made of straw circles joined together in the form shown in the figure below: 
    o o o o o o o
    o x x x x x o
    o x x x x x o
    o x x x x x o
    o o o o o o o
All the circles on the edges of each mat were white and the inner circles red. I noted that there were 20 white circles and 15 red circles.

I wondered if it were possible to make a mat in this form where the number of white circles equaled the number of red circles. I found there were two possible solutions. Can you find them and show that there are only two such solutions?

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