1. **State the problem:** Simplify the expression $$\frac{x^2 y^2 + 3xy}{4x^2 - 1} \div \frac{xy + 3}{2x + 1}$$. 2. **Rewrite division as multiplication:** Dividing by a fraction is the same as multiplying by its reciprocal. So, $$\frac{x^2 y^2 + 3xy}{4x^2 - 1} \times \frac{2x + 1}{xy + 3}$$. 3. **Factor where possible:** - Numerator of first fraction: $$x^2 y^2 + 3xy = xy(xy + 3)$$. - Denominator of first fraction: $$4x^2 - 1 = (2x - 1)(2x + 1)$$ (difference of squares). 4. **Rewrite the expression with factors:** $$\frac{xy(xy + 3)}{(2x - 1)(2x + 1)} \times \frac{2x + 1}{xy + 3}$$. 5. **Cancel common factors:** - $$xy + 3$$ appears in numerator and denominator. - $$2x + 1$$ appears in numerator and denominator. After cancellation, we get: $$\frac{xy}{2x - 1}$$. 6. **Final simplified expression:** $$\boxed{\frac{xy}{2x - 1}}$$. This is the simplified form of the original expression.