If $|\overrightarrow {\rm{a}} \times \overrightarrow {\rm{b}} {|^2} + |\overrightarrow {\rm{a}} \cdot \overrightarrow {\rm{b}} {|^2} = 144$ and $|\overrightarrow {\rm{a}} | = 4,$ then $|\overrightarrow {\rm{b}} |$ is equal to

$|\overrightarrow {\rm{a}} \times \overrightarrow {\rm{b}} {|^2} + |\overrightarrow {\rm{a}} \cdot \overrightarrow {\rm{b}} {|^2}$

$= 144 = |\overrightarrow {\rm{a}} {|^2} \cdot |\overrightarrow {\rm{b}} {|^2}$

$\Rightarrow$ $|\overrightarrow {\rm{a}} {|^2}|\overrightarrow {\rm{b}} {|^2} = 144$

$\Rightarrow$ $|\overrightarrow {\rm{b}} {|^2} = \frac{{144}}{{|\overrightarrow {\rm{a}} {|^2}}} = \frac{{144}}{{16}} = 9$

$\therefore$ $|\overrightarrow {\rm{b}} | = 3$

As we know, $|\overrightarrow {\rm{a}} \times \overrightarrow {\rm{b}} {|^2} + |\overrightarrow {\rm{a}} \cdot \overrightarrow {\rm{a}} {|^2}$

$= |\overrightarrow {\rm{a}} {|^2}|\overrightarrow {\rm{b}} {|^2}$.