Answer
$${\bf{w}} = \frac{{12}}{5}{\bf{i}} + \frac{{16}}{5}{\bf{j}}$$
Work Step by Step
$$\eqalign{
& {\text{Let }}{\bf{u}} = 4{\bf{i}} + 2{\bf{j}},{\text{ }}{\bf{v}} = 3{\bf{i}} + 4{\bf{j}} \cr
& {\text{Let }}{\bf{w}} = {\text{pro}}{{\text{j}}_{\bf{v}}}{\bf{u}} = \left( {\frac{{{\bf{u}} \cdot {\bf{v}}}}{{{{\left\| {\bf{v}} \right\|}^2}}}} \right){\bf{v}} \cr
& {\bf{w}} = \left( {\frac{{\left( {4{\bf{i}} + 2{\bf{j}}} \right) \cdot \left( {3{\bf{i}} + 4{\bf{j}}} \right)}}{{{{\left\| {3{\bf{i}} + 4{\bf{j}}} \right\|}^2}}}} \right)\left( {3{\bf{i}} + 4{\bf{j}}} \right) \cr
& {\bf{w}} = \left( {\frac{{12 + 8}}{{9 + 16}}} \right)\left( {3{\bf{i}} + 4{\bf{j}}} \right) \cr
& {\bf{w}} = \frac{4}{5}\left( {3{\bf{i}} + 4{\bf{j}}} \right) \cr
& {\bf{w}} = \frac{{12}}{5}{\bf{i}} + \frac{{16}}{5}{\bf{j}} \cr} $$
