question of chi-squared distribution

#1
Hi

I would like to show that the pooled standard deviation of the sum of 2 independent normally distributed variables is a biased estimator, and to show the degree of bias

i have shown my working so far in the attachment.

I would appreciate help on how some algebraic manipulation of the d.f of the chi-squared function to obtain a value for E(S) without any x terms.

Thanks very much
Preeya