1. **State the problem:** Simplify or understand the expression $$(x^3 + 2x)^{37}$$. 2. **Formula and rules:** This is a binomial expression raised to a power. The binomial theorem states: $$ (a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k $$ where $\binom{n}{k}$ is the binomial coefficient. 3. **Apply the binomial theorem:** Here, $a = x^3$ and $b = 2x$, and $n = 37$. 4. **General term:** The $k$-th term in the expansion is: $$ T_{k+1} = \binom{37}{k} (x^3)^{37-k} (2x)^k = \binom{37}{k} 2^k x^{3(37-k)} x^k = \binom{37}{k} 2^k x^{111 - 3k + k} = \binom{37}{k} 2^k x^{111 - 2k} $$ 5. **Interpretation:** The full expansion is a sum of terms of the form above for $k=0$ to $37$. 6. **Summary:** The expression expands to $$ \sum_{k=0}^{37} \binom{37}{k} 2^k x^{111 - 2k} $$ which is a polynomial in $x$ with decreasing powers by 2 each term. This is the simplified form using the binomial theorem; fully expanding would be lengthy but follows this pattern.