1. **State the problem:** Write the expression $$\frac{5x - 2}{3} + \frac{2}{5x + 2}$$ as a single fraction in simplest form. 2. **Find a common denominator:** The denominators are 3 and $$5x + 2$$. The common denominator is $$3(5x + 2)$$. 3. **Rewrite each fraction with the common denominator:** $$\frac{5x - 2}{3} = \frac{(5x - 2)(5x + 2)}{3(5x + 2)}$$ $$\frac{2}{5x + 2} = \frac{2 \cdot 3}{3(5x + 2)} = \frac{6}{3(5x + 2)}$$ 4. **Add the numerators:** $$\frac{(5x - 2)(5x + 2) + 6}{3(5x + 2)}$$ 5. **Expand the numerator:** Use the difference of squares formula: $$(5x - 2)(5x + 2) = (5x)^2 - 2^2 = 25x^2 - 4$$ 6. **Substitute back:** $$\frac{25x^2 - 4 + 6}{3(5x + 2)} = \frac{25x^2 + 2}{3(5x + 2)}$$ 7. **Check for simplification:** The numerator $$25x^2 + 2$$ does not factor nicely, so this is the simplest form. **Final answer:** $$\frac{25x^2 + 2}{3(5x + 2)}$$