1. **State the problem:** Simplify the expression $$\frac{x^2 - 4}{2} + \frac{x^2 + 2x}{4}$$ as a single fraction in simplest form. 2. **Find a common denominator:** The denominators are 2 and 4. The least common denominator (LCD) is 4. 3. **Rewrite each fraction with the LCD:** $$\frac{x^2 - 4}{2} = \frac{2(x^2 - 4)}{2 \times 2} = \frac{2(x^2 - 4)}{4}$$ 4. **Rewrite the expression:** $$\frac{2(x^2 - 4)}{4} + \frac{x^2 + 2x}{4}$$ 5. **Combine the numerators over the common denominator:** $$\frac{2(x^2 - 4) + (x^2 + 2x)}{4}$$ 6. **Expand and simplify the numerator:** $$2(x^2 - 4) + (x^2 + 2x) = 2x^2 - 8 + x^2 + 2x = 3x^2 + 2x - 8$$ 7. **So the expression is:** $$\frac{3x^2 + 2x - 8}{4}$$ 8. **Factor the numerator if possible:** Try to factor $$3x^2 + 2x - 8$$. Find two numbers that multiply to $$3 \times (-8) = -24$$ and add to 2. These numbers are 6 and -4. Rewrite the middle term: $$3x^2 + 6x - 4x - 8$$ Group terms: $$3x(x + 2) - 4(x + 2)$$ Factor out common binomial: $$(3x - 4)(x + 2)$$ 9. **Final simplified expression:** $$\frac{(3x - 4)(x + 2)}{4}$$ This is the expression as a single fraction in simplest form.