There certainly is a better way then expanding the entire expression, it is to use the binomial theorem. You need to use the binomial coefficients which you may have seen before. For integers n and k, with k ≤ n, the binomial coefficients are given by.
I wrote the notation nCk since some high school texts use this notation but I am going to use since this is the notation mathematicians use. Using this notation the binomial theorem states.
For your example you have n = 7, a = 2x and b = -y.
Hence, for example the fifth term is