$\text{I.}{\left(\frac{3}{4}m-4n\right)}^2={\left(\frac{3}{4}m\right)}^2-2\cdot \frac{3}{4}m\cdot 4n+{\left(4n\right)}^2=\frac{9}{16}{m}^2-6mn+16n^2=16n^2+\frac{9}{16}{m}^2-6mn\text{C.}$   

$\text{II.}\left(-2m+\frac{3}{4}n\right)\cdot \left(\frac{3}{4}n-2m\right)=\left(\frac{3}{4}n-2m\right)\left(\frac{3}{4}n-2m\right)={\left(\frac{3}{4}n-2m\right)}^2={\left(\frac{3}{4}n\right)}^2-2\cdot \frac{3}{4}n\cdot 2m+{\left(2m\right)}^2=\frac{9}{16}{n}^2-3n+4m^2\text{D.}$  

$\text{III.}\left(\frac{3}{4}m-2n\right)\left(\frac{3}{4}m-2n\right)={\left(\frac{3}{4}m-2n\right)}^2={\left(\frac{3}{4}m\right)}^2-2\cdot \frac{3}{4}m\cdot 2n+{\left(2n\right)}^2=\frac{9}{16}{m}^2-3mn+4n^2\text{B.}$  

$\text{IV.}{\left(\frac{3}{4}n-4m\right)}^2={\left(\frac{3}{4}n\right)}^2+2\cdot \frac{3}{4}n\cdot 4m+{\left(4m\right)}^2=\frac{9}{16}{n}^2-6mn+16m^2\text{A.}$