Answer
$\textbf a+ \textbf b = 5\textbf i - \textbf j$
$2\textbf a+ 3\textbf b = 11\textbf i -4 \textbf j$
$|\textbf a| = \sqrt {17}$
$|\textbf {a-b}| = 3\sqrt {2}$
Work Step by Step
$\textbf a+ \textbf b = (4\textbf i+ \textbf j) + (\textbf i - 2\textbf j) = 5\textbf i - \textbf j$
$2\textbf a+ 3\textbf b = 2 (4\textbf i+ \textbf j) + 3(\textbf i - 2\textbf j) = (8\textbf i+ 2\textbf j) + (3\textbf i - 6\textbf j) = 11\textbf i -4 \textbf j$
$|\textbf a| = \sqrt {a_1^2 + a_2^2} = \sqrt {4^2 + 1^2} = \sqrt {16 + 1} = \sqrt {17}$
$|\textbf {a-b}| = \sqrt {(a_1 - b_1)^2 + (a_2 - b_2)^2} = \sqrt {(4-1)^2 + (1 -(- 2))^2} = \sqrt {3^2 + 3^2} = \sqrt {9 + 9} = \sqrt {18} = 3\sqrt {2}$