a^2 + b / a^2 - b + d^2 - b / a^2 + b - a^4 + b^4 / a^4 - b^2.

Steps to Solve

1. Identify the Expression Components

The given expression combines several rational expressions involving variables (a), (b), and (d): $$\frac{a^2 + b}{a^2 - b} + \frac{d^2 - b}{a^2 + b} - \frac{a^4 + b^4}{a^4 - b^2}$$

2. Find a Common Denominator for the First Two Fractions

The least common denominator (LCD) for the first two fractions is ((a^2 - b)(a^2 + b)). To rewrite each fraction:

For the first term: $$\frac{(a^2 + b)(a^2 + b)}{(a^2 - b)(a^2 + b)} = \frac{(a^2 + b)^2}{(a^2 - b)(a^2 + b)}$$

For the second term: $$\frac{(d^2 - b)(a^2 - b)}{(a^2 + b)(a^2 - b)} = \frac{(d^2 - b)(a^2 - b)}{(a^2 + b)(a^2 - b)}$$

3. Combine the First Two Fractions

Combine the first two fractions using the common denominator: $$\frac{(a^2 + b)^2 + (d^2 - b)(a^2 - b)}{(a^2 + b)(a^2 - b)}$$

4. Simplify the Numerator

Expand the numerator:

• Expanding ((a^2 + b)^2) gives (a^4 + 2a^2b + b^2).
• The other part becomes ((d^2a^2 - d^2b - ba^2 + b^2)).

The entire numerator now reads: $$a^4 + 2a^2b + b^2 + d^2a^2 - d^2b - ba^2 + b^2$$

Combine like terms.

5. Denominator of the Third Fraction

The denominator from the third fraction is (a^4 - b^2). The third fraction can be simplified as: $$-\frac{a^4 + b^4}{a^4 - b^2}$$

6. Rewrite the Whole Expression

Now express the entire equation: Combine the simplified numerator from step 4 and the negative of the third fraction: $$\frac{(\text{simplified numerator from step 4})}{(a^2 + b)(a^2 - b)} - \frac{(a^4 + b^4)}{(a^4 - b^2)}$$

7. Combine All Terms

At this point, attempt to combine all fractions together by finding a common denominator between the two parts. After finding the common denominator, add or subtract as necessary.

8. Final Simplification

Finally, simplify the entire fraction, if necessary, and check for any possible cancellations.