1. **State the problem:** We want to expand and simplify the expression $$(a + b)^4$$. 2. **Formula used:** The binomial theorem states that $$(a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k$$ where $\binom{n}{k}$ are binomial coefficients. 3. **Apply the formula for $n=4$:** $$ (a + b)^4 = \binom{4}{0}a^4b^0 + \binom{4}{1}a^3b^1 + \binom{4}{2}a^2b^2 + \binom{4}{3}a^1b^3 + \binom{4}{4}a^0b^4 $$ 4. **Calculate binomial coefficients:** $$ \binom{4}{0} = 1, \quad \binom{4}{1} = 4, \quad \binom{4}{2} = 6, \quad \binom{4}{3} = 4, \quad \binom{4}{4} = 1 $$ 5. **Substitute coefficients and simplify:** $$ (a + b)^4 = 1 \cdot a^4 + 4 \cdot a^3 b + 6 \cdot a^2 b^2 + 4 \cdot a b^3 + 1 \cdot b^4 $$ 6. **Final expanded form:** $$ (a + b)^4 = a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4 $$ This is the fully expanded and simplified form of $$(a + b)^4$$.