Answer: D

Small Hint:

In vector notation, $A'=2A-D,$ $B'=2B-A,$ and similarly for the other two vertices

Big Hint:

Substitute these expressions into the cross-product area formula for a polygon

Solution:

Using position vectors, the equal extensions give $$\begin{aligned} A'&=2A-D,& B'&=2B-A,\\ C'&=2C-B,& D'&=2D-C. \end{aligned}$$ Substitution into the oriented polygon-area sum shows that the diagonal cross terms cancel: $$\begin{aligned} &A'\mathbin{\times}B' +B'\mathbin{\times}C'\\ &\quad+C'\mathbin{\times}D' +D'\mathbin{\times}A'\\ &=5A\mathbin{\times}B +5B\mathbin{\times}C\\ &\quad+5C\mathbin{\times}D +5D\mathbin{\times}A. \end{aligned}$$ Hence the outer area is five times the original area, or $5\cdot10=50.$

Therefore, the correct answer is D.