Answer:
The coeff. of x^4 in this expansion is -1280.
Explanation:
Start by writing down a Pascal's Triangle:
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
Example: write out (x + y)^3:
Use the 4th row of Pascal's Triangle: 1(x^3) + 3(x^2) + 3(x^1) + 1. Note how
the powers of x decrease from 3 through 2, 1 and 0.
Now let's apply this to the problem at hand. Use the coefficients in the 6th row of the Triangle, above:
1[4x]^5 + 5[4x]^4·(-1) + ....
The first term is 1[4x]^5, or [4x]^5, or 4^5·x^5, or 1024·x^5.
The second term is 5[4x]^4·(-1), or 5·4^4·(-1), or 5[256](-1) = -1280.
Thus, the coeff. of x^4 in this expansion is -1280.