I want to show that $G=GL(n,\mathbb R)\times \mathbb R^n$ where $\mu:G\times G \rightarrow G$ is defined by $\mu((x,v),(y,w))=(xy,y^{-1}v+w)$ is a Lie group.
Clearly $G$ is a manifold since it is the product of two manifolds. It remains to show that $\mu$ is smooth with respect to this manifold structure.
It makes sense that $\mu$ is smooth. Informally my argument would be in $GL(n,R)$ the map $\mu_1(x,y)=xy$ is smooth. In $\mathbb R^n$ the map $\mu_2(v,w)=Av+w$ is smooth for any matrix $A$. But I'm not sure how to show that this gives that $\mu$ is smooth. Could it not be that as $A$ changes $\mu_2$ does not vary smoothly?
All the coordinates of this map are rational functions with non-nanishing denominator (use $y^{-1}=\frac{1}{\det(y)}^t\mathrm{com}(y)$), so the map itself is indeed smooth.