Let's simplify the given expression step by step.

We are given the following complex fraction:

$\frac{\frac{b^2}{a+b}}{\frac{a^2 + 1}{a+b}}.$

Step 1: Simplify the overall fraction.

When you have a fraction divided by another fraction, you multiply by the reciprocal of the denominator. So:

$\frac{\frac{b^2}{a+b}}{\frac{a^2 + 1}{a+b}} = \frac{b^2}{a+b} \times \frac{a+b}{a^2 + 1}.$

Step 2: Cancel out common terms.

Since both fractions have $a + b$ in the numerator and denominator, we can cancel them out:

$\frac{b^2}{a^2 + 1}.$

Thus, the simplified expression is:

$\frac{b^2}{a^2 + 1}.$

This matches one of the choices in the options. The correct answer is:

$\boxed{\frac{b^2}{a^2 + 1}}.$

Would you like further details or have any questions?

Here are 5 related questions:

1. How would you simplify a more complex fraction with multiple terms?
2. What happens when both the numerator and denominator contain square roots?
3. How do you solve algebraic expressions involving fractional exponents?
4. Can you solve similar expressions if the variables were different?
5. How do you manage simplifying when variables are inside trigonometric functions?

Tip: Always look for common factors to cancel out when simplifying complex fractions!