Divide polynomials with exponents and simplify The expression is

$$\frac{p^2q^2}{m^2-n^2} \cdot \frac{m^2+2mn+n^2}{3p^2+2pq-q^2}.$$

How could I divide and simplify this?

We can write the product of two rational fractions as one fraction, and use commutativity of multiplication to rearrange factors, as needed:
$$\frac{p^2q^2}{m^2-n^2} \cdot \frac{m^2+2mn+n^2}{3p^2+2pq-q^2} = \frac{p^2q^2(m^2 + 2mn+n^2)}{(3p^2 + 2pq - q^2)(m^2 - n^2)}$$
Here, you have quadratics in each of the numerator and denominator that can be nicely factored. 
$$\frac{p^2q^2(\overbrace{m^2 + 2mn+n^2}^{(m+n)^2})}{(3p^2 + 2pq - q^2)(m^2 - n^2)} = \frac{p^2q^2\color{blue}{\bf (m + n)}(m+n)}{(p+q)(3p-q)\color{blue}{\bf (m+n)}(m-n)}$$
We can cancel the common factor appearing  in each of the numerator and denominator, which gives us:$$\frac{p^2q^2(m+n)}{(p+q)(3p-q)(m-n)}$$
This is about as simplified as it gets.
