Sum of two independent normal distributed random variables
Use moment generating function $$M_{X_1+X_2}(t)=\exp(\sigma_1^2t^2/2+\sigma_2^2t^2/2)=\exp((\sigma_1^2+\sigma_2^2)t^2/2)$$ and so $X_1+X_2\sim N(0,\sigma_1^2+\sigma_2^2)$.
Sum of two independent normal distributed random variables
Use moment generating function $$M_{X_1+X_2}(t)=\exp(\sigma_1^2t^2/2+\sigma_2^2t^2/2)=\exp((\sigma_1^2+\sigma_2^2)t^2/2)$$ and so $X_1+X_2\sim N(0,\sigma_1^2+\sigma_2^2)$.