Let’s solve the given mathematical expression step by step. The expression is:

$\frac{a^2 + 2ab + b^2}{a + b} \div \left(\frac{1}{a} + \frac{1}{b}\right)$

Step 1: Simplify $\frac{a^2 + 2ab + b^2}{a + b}$

Notice that $a^2 + 2ab + b^2$ is a perfect square trinomial, so it can be factored as:

$a^2 + 2ab + b^2 = (a + b)^2$

Substituting this into the fraction:

$\frac{a^2 + 2ab + b^2}{a + b} = \frac{(a + b)^2}{a + b}$

Simplify the fraction by canceling one $a + b$ (provided $a + b \neq 0$):

$\frac{a^2 + 2ab + b^2}{a + b} = a + b$

Step 2: Simplify $\frac{1}{a} + \frac{1}{b}$

Using the common denominator $ab$, we combine the terms:

$\frac{1}{a} + \frac{1}{b} = \frac{b}{ab} + \frac{a}{ab} = \frac{a + b}{ab}$

Step 3: Rewrite the division

Now substitute the simplified expressions into the original problem:

$(a + b) \div \frac{a + b}{ab}$

Division by a fraction is the same as multiplying by its reciprocal:

$(a + b) \cdot \frac{ab}{a + b}$

Step 4: Simplify the product

Cancel the $a + b$ terms (provided $a + b \neq 0$):

$(a + b) \cdot \frac{ab}{a + b} = ab$

Final Answer:

$ab$

Would you like me to explain any specific step in more detail or provide additional examples?

Related Questions:

1. What happens if $a + b = 0$ in the original expression?
2. How can you generalize this expression for higher powers, like $(a^3 + 3a^2b + 3ab^2 + b^3)$?
3. How does this problem relate to factoring and simplifying rational expressions?
4. Can this solution be verified by substituting specific values for $a$ and $b$?
5. How does dividing by a fraction compare to multiplying fractions?

Tip:

When simplifying algebraic fractions, always look for factoring opportunities to cancel terms effectively.