a ^ 2 + 2ab/a ^ 2 : a ^ 2 + 4ab + 4b ^ 2/ab

For the first question, let's simplify the expression:

$\frac{a^2 + 2ab}{a^2} : \frac{a^2 + 4ab + 4b^2}{ab}$

To simplify this expression, we can rewrite the division as multiplication by the reciprocal:

$\frac{a^2 + 2ab}{a^2} \cdot \frac{ab}{a^2 + 4ab + 4b^2}$

Now, let's simplify the numerator and denominator separately:

Numerator:
$a^2 + 2ab$

Denominator:
$a^2 + 4ab + 4b^2$

Now, let's substitute these simplified expressions back into the original expression:

$\frac{a^2 + 2ab}{a^2} \cdot \frac{ab}{a^2 + 4ab + 4b^2}$

$= \frac{(a^2 + 2ab) \cdot ab}{a^2 \cdot (a^2 + 4ab + 4b^2)}$

$= \frac{a^3b + 2a^2b^2}{a^4 + 4a^3b + 4a^2b^2}$

Therefore, the simplified expression is $\frac{a^3b + 2a^2b^2}{a^4 + 4a^3b + 4a^2b^2}$.