In order to find the appropriate coefficients for the binomila expansion in the case of such high exponent, it is convenient to use Pascal's triangle .
I am pasting an image of a Pascal triangle up to the power 7 below:
Notice that the coefficients are lasted from left to right as the first one with power 7 for the 2x term, and no inclusion of the term y, and then decreasing the power of the 2x term, as the term in y increases power.
That is, the terms ordered from highest in 2x to lower go like:
(2x)^7 + 7 (2x)^6 (y) + 21 (2x)^5 (y)^2 + 35 (2x)^4 (y)^3 + ...
We stop here since we are asked specifically for the fourth term, which is:
35 (2x)^4 (y)^3 = 35 * 16 x^4 y^3 = 560 x^4 y^3