1. **State the problem:** Add the fractions $\frac{x+4}{x^2+2}$ and $\frac{x^2-2}{x-8}$ and simplify the result. 2. **Find the common denominator:** The denominators are $x^2+2$ and $x-8$. The common denominator is their product: $$ (x^2+2)(x-8) $$ 3. **Rewrite each fraction with the common denominator:** $$ \frac{x+4}{x^2+2} = \frac{(x+4)(x-8)}{(x^2+2)(x-8)} $$ $$ \frac{x^2-2}{x-8} = \frac{(x^2-2)(x^2+2)}{(x-8)(x^2+2)} $$ 4. **Add the numerators:** $$ (x+4)(x-8) + (x^2-2)(x^2+2) $$ 5. **Expand each product:** $$ (x+4)(x-8) = x^2 - 8x + 4x - 32 = x^2 - 4x - 32 $$ $$ (x^2-2)(x^2+2) = x^4 + 2x^2 - 2x^2 - 4 = x^4 - 4 $$ 6. **Sum the expanded numerators:** $$ x^2 - 4x - 32 + x^4 - 4 = x^4 + x^2 - 4x - 36 $$ 7. **Write the combined fraction:** $$ \frac{x^4 + x^2 - 4x - 36}{(x^2+2)(x-8)} $$ 8. **Check if numerator can be factored:** Try grouping: $$ (x^4 + x^2) - (4x + 36) = x^2(x^2 + 1) - 4(x + 9) $$ No common factor to factor further easily. 9. **Check denominator expansion:** $$ (x^2+2)(x-8) = x^3 - 8x^2 + 2x - 16 $$ 10. **Final simplified expression:** $$ \frac{x^4 + x^2 - 4x - 36}{x^3 - 8x^2 + 2x - 16} $$ 11. **Match with given options:** None of the options exactly match this form, but option A is: $$ \frac{x^3 + 4x^2 - 2x - 8}{x^3 - 8x^2 + 2x - 16} $$ which is different. Option E is: $$ \frac{x^2 + 12x - 32}{x^4 + 4x^2 - 4} $$ which is also different. Option D is: $$ x^2 - 4x + \frac{-32}{x^4 - 4} $$ which is not a single fraction. Option B and C are simple fractions. Therefore, the correct simplified sum is: $$ \frac{x^4 + x^2 - 4x - 36}{x^3 - 8x^2 + 2x - 16} $$ **Answer:** None of the provided options exactly match the simplified sum. **Slug:** add fractions **Subject:** algebra