$$\overrightarrow a = \alpha \widehat i + \widehat j + \beta \widehat k$$, $$\overrightarrow b = 3\widehat i - 5\widehat j + 4\widehat k$$

$$\overrightarrow a \times \overrightarrow b = - \widehat i + 9\widehat j + 12\widehat k$$

$$\left| {\matrix{ {\widehat i} & {\widehat j} & {\widehat k} \cr \alpha & 1 & \beta \cr 3 & { - 5} & 4 \cr } } \right| = - \widehat i + 9\widehat j + 12\widehat k$$

$$4 + 5\beta = - 1 \Rightarrow \beta = - 1$$

$$ - 5\alpha - 3 = 12 \Rightarrow \alpha = - 3$$

$$\overrightarrow b - 2\overrightarrow a = 3\widehat i - 5\widehat j + 4\widehat k - 2\left( { - 3\widehat i + \widehat j - \widehat k} \right)$$

$$\overrightarrow b - 2\overrightarrow a = 9\widehat i - 7\widehat j + 6\widehat k$$

$$\overrightarrow b + \overrightarrow a = \left( {3\widehat i - 5\widehat j + 4\widehat k} \right) + \left( { - 3\widehat i + \widehat j - \widehat k} \right)$$

$$\overrightarrow b + \overrightarrow a = - 4\widehat j + 3\widehat k$$

Projection of $$\overrightarrow b - 2\overrightarrow a $$ on $$\overrightarrow b + \overrightarrow a $$ is $$ = {{\left( {\overrightarrow b - 2\overrightarrow a } \right)\,.\,\left( {\overrightarrow b + \overrightarrow a } \right)} \over {\left| {\overrightarrow b + \overrightarrow a } \right|}}$$

$$ = {{28 + 18} \over 5} = {{46} \over 5}$$