Official solution
Given mean (\[\mu \]) = 8
variance (\[{\sigma ^2}\]) = 16
No of observations (N) = 7
Let the two unknown observation = x and y
We know,
\[{\sigma ^2} = {{\sum {x_i^2} } \over N} - {\mu ^2}\] = 16
\[ \Rightarrow \] \[{{{2^2} + {4^2} + {{10}^2} + {{12}^2} + {{14}^2} + {x^2} + {y^2}} \over 7} - {\left( 8 \right)^2}\] = 16
\[ \Rightarrow \] x2 + y2 = 100 ........(1)
We know,
\[\mu \] = \[{{\sum {{x_i}} } \over N}\] = 8
\[ \Rightarrow \] \[{{2 + 4 + 10 + 12 + 14 + x + y} \over 7}\] = 8
\[ \Rightarrow \] x + y = 14 ...........(2)
As (x + y)2 = x2 + y2 + 2xy
\[ \Rightarrow \] (14)2 = 100 + 2xy
\[ \Rightarrow \] 196 = 100 + 2xy
\[ \Rightarrow \] xy = 48