It's imperative to note the following: A random variable that follows the chi-square distribution has a mean of n and a variance of 2n, where n is the number of degrees of freedom. We could approach the question from two different perspectives:

i. Summation of independent variables
Since n = 2, $E(X_i) = 2$, and $\text{Var}(X_i) = 4$
$X_1$ and $X_2$ are independent, which means:
$E(X_1 + X_2) = 4$, and $\text{Var}(X_1 + X_2) = 8$

Therefore, for the total loss from the two projects,
Mean = 4 * $100,000 = $400,000 while Variance = 8 * $100,000 = $800,000

ii. Summation of two chi-square random variables
If A and B are two chi-square random variables with m and n degrees of freedom, respectively, then the sum of A and B is ALSO a chi-square variable with m + n degrees of freedom.

Therefore,
$\chi^2_2 + \chi^2_2 = \chi^2_4$

Working out the problem with this result gives the same values as above.