Simplify. 31. \( \frac{a^{4}-b^{4}}{b^{2}-a^{2}} \)

To simplify the expression \( \frac{a^{4}-b^{4}}{b^{2}-a^{2}} \), we can utilize the difference of squares. The numerator \( a^{4}-b^{4} \) can be factored as \( (a^{2}-b^{2})(a^{2}+b^{2}) \). The denominator \( b^{2}-a^{2} \) can be factored as \( -(a^{2}-b^{2}) \). Now substituting those factorizations, we have: \[ \frac{(a^{2}-b^{2})(a^{2}+b^{2})}{-(a^{2}-b^{2})} \] We can cancel \( a^{2}-b^{2} \) from the numerator and the denominator (assuming \( a^{2} \neq b^{2} \)), leading us to: \[ -(a^{2}+b^{2}) \] So the simplified result is \( -(a^{2}+b^{2}) \).