Solution:

From the 2nd equation, we have $a^{2} + b^{2} = \frac{9}{4} + 2 a b$.

Using this for the 1st equation, we have
$$\frac{a^{2} + b^{2}}{a b} = \frac{\frac{9}{4} + 2 a b}{a b} = \frac{5}{2}.$$
It can be solved that $a b = \frac{9}{2}$.

Moreover, from the 2nd equation, we have
$$(a + b)^{2} = \frac{9}{4} + 4 a b = \frac{81}{4}.$$
Thus, $a + b = \frac{9}{2}$ or $-\frac{9}{2}$.

And
$$a^{2} + 2 a b + b^{2} + 2 a^{2} b + 2 a b^{2} + a^{2} b^{2} = (a b + a + b)^{2} = 81 \text{ or } 0.$$