$$\sqrt{2x_1^2+x_1x_2+4x_2^2+4}+\frac{(x_1-x_2+x_3+1)^2}{x_1+x_2}\leq6$$
I think the square root part can be written as follows:
$$||(\frac12(x_1+x_2)^2,\frac{\sqrt3}{\sqrt2}x_1,\frac{\sqrt7}{\sqrt2}x_2,2)^T||$$
where ||.|| is Euclidian norm. It is a composition of a 2-norm with an affine transformation and therefore is convex.
But stuck on the second expression. I know it can be found by Hessian, but for more complicated functions it can be very costly in terms of time and it requires a lot of attention if you solve it by hand. I wanted to ask if there is relatively shorter way to do that as I have mentioned for the square root expression.
How do you think I can show it is convex?
Edit: Also looking for this one:
$$\frac{(x_1^2+x_2^2)^2}{(x_1+x_2)^2}$$