The expression $\frac{a^2 + b^2}{a - b}$ can be simplified or rewritten in different forms depending on the situation, but by itself, it doesn't simplify to a basic algebraic identity.

However, it can be factored or rewritten in different ways in some cases. For instance, if you wanted to manipulate this expression algebraically, one possible approach is using the difference of squares or adding and subtracting $ab$ in the numerator, but the expression doesn't have an immediate, simple factorization like common identities (e.g., $a^2 - b^2 = (a - b)(a + b)$).

One possibility is to add and subtract $ab$ in the numerator:

$\frac{a^2 + b^2}{a - b} = \frac{(a^2 - b^2) + 2b^2}{a - b}$ Then, the term $a^2 - b^2$ can be factored as: $= \frac{(a - b)(a + b) + 2b^2}{a - b}$

Now, you can separate the terms: $= (a + b) + \frac{2b^2}{a - b}$ So, the expression becomes: $(a + b) + \frac{2b^2}{a - b}$

This is one way to rewrite the expression, but in general, the original form doesn't have a simple algebraic identity that simplifies it further without additional context or constraints.

Would you like more details on any specific step or another approach to this problem?

Here are some related questions:

1. How do you factor expressions like $a^2 - b^2$?
2. What is the difference of squares formula, and how is it applied?
3. How can you simplify expressions with fractions in the numerator and denominator?
4. What happens if you set $a = b$ in this expression?
5. How do you handle algebraic expressions involving both squares and linear terms?

Tip: When simplifying algebraic expressions, always check for common factorizations like the difference of squares, perfect square trinomials, or sum and difference of cubes.