Answer
$x^{4}$ + $4x^{3}y$ + $6x^{2}y^{2}$+ $4xy^{4}$+ $y^{4}$
Work Step by Step
$a)(x+y)^{4}$ = (x + y)(x + y)(x + y)(x + y). To obtain a term of the form $x^4$, an x must be chosen in each of the sums, and this can be done in only one way. Thus, the $x^{4}$ term in the product has a coefficient of 1. To obtain a term of the form $x^3y$, an x must be chosen in 3 of the four sums (and consequently a y in the other sum). Hence, the number of such terms is the number (4C3). Then you must follow the same ideology for the rest of the terms to obtain your coefficients for the final answer.
b) To obtain the answer, you must use the binomial theorem. Thus $(x+y)^4 = \sum_{n=0}^{4} (4Cn) x^{4-n}y^n$.
