There is no quick answer unless you use this.
Before we had that, we had to use this: Binomial theorem - Wikipedia, the free encyclopedia
So , and
Picking out the terms gives:
, so the coefficient we're looking for is 40.
By observation we can see that the first expansion will have three terms, one will be constant 1, second with x ^2 and third x^4.
The second expansion will have powers of x from 0 to 8.
Since we have to multiply and get coefficient of x^3, we need to know first two terms of first expansion i.e., 1 -2x^2 and from the second expansion we need to know terms containing x, and x^3 that is 2nd and fourth term.
Thus by knowing four terms we can multiply and get the coefficient of x^3.