1. We are asked to expand the binomial expression $(x + y)^4$. 2. The binomial theorem states: $$ (a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k $$ where $\binom{n}{k} = \frac{n!}{k!(n-k)!}$ is the binomial coefficient. 3. For $(x + y)^4$, $a = x$, $b = y$, and $n = 4$. 4. Calculate each term: - $k=0$: $\binom{4}{0} x^{4} y^{0} = 1 \cdot x^{4} \cdot 1 = x^{4}$ - $k=1$: $\binom{4}{1} x^{3} y^{1} = 4 \cdot x^{3} \cdot y = 4x^{3}y$ - $k=2$: $\binom{4}{2} x^{2} y^{2} = 6 \cdot x^{2} \cdot y^{2} = 6x^{2}y^{2}$ - $k=3$: $\binom{4}{3} x^{1} y^{3} = 4 \cdot x \cdot y^{3} = 4xy^{3}$ - $k=4$: $\binom{4}{4} x^{0} y^{4} = 1 \cdot 1 \cdot y^{4} = y^{4}$ 5. Combine all terms: $$ (x + y)^4 = x^{4} + 4x^{3}y + 6x^{2}y^{2} + 4xy^{3} + y^{4} $$ This is the expanded form of $(x + y)^4$.