$$f(x,y,z)=4x+2y+z$$ $$ E=\{(x,y,z) \in R : (x+1)^2+4y^2+4z^2=4\}$$
I know I should write here what I already did but I could come up with literally nothing. Should I just find extreme values of $g(x,y,z):=4x+2y+z-((x+1)^2+4y^2+4z^2-4)$ does it even make sense?
Hint (without Lagrange multipliers)
Write $x+1 = 2 \cos\theta$ and $y = \sin\theta \cos\varphi$ and $z=\sin\theta\sin\varphi$. You are left with finding the extrema of a smooth periodic function in $\mathbb{R}^2$.