To solve the equation $3^{2x + 3} = 5^{3x + 1}$, we can take the logarithm of both sides to bring down the exponents. Here's the step-by-step process:

1. Take the logarithm of both sides (using natural logarithm $\ln$ or common logarithm $\log$): $\ln(3^{2x + 3}) = \ln(5^{3x + 1})$

2. Use the logarithm power rule $\ln(a^b) = b\ln(a)$ to bring down the exponents: $(2x + 3)\ln(3) = (3x + 1)\ln(5)$

3. Expand both sides: $2x\ln(3) + 3\ln(3) = 3x\ln(5) + \ln(5)$

4. Group the terms with $x$ on one side and