How to open $[1 + \langle x,y\rangle]^2$

Question

I'm trying to prove something, and I got stuck in some algebraic stuff. I'm trying to understand this answer, which he is doing there:

$[1 + \langle x,y\rangle]^2$ = $\langle[1, \sqrt{2}y_1, y_1^2, \sqrt{2}y_2, y_2^2, \sqrt{2}y_1y_2], [1, \sqrt{2}x_1, x_1^2, \sqrt{2}x_2, x_2^2, \sqrt{2}x_1x_2]^T\rangle$

I can't figure out how to open this expression like he did. If anyone can explain, or give a source that gives a full example that would be amazing.

I'm not sure about the tags, so please edit it needed.

Answer 1

$$ [1+\langle x,y\rangle]^2= [1+x_1 y_1 +x_2 y_2]^2= 1+2x_1 y_1 +(x_1 y_1)^2 +2 x_2 y_2 +(x_2 y_2)^2 +2 x_1 y_1 x_2 y_2 $$ which is pretty clearly the inner product in $\mathbb{R}^6$ of the two given vectors.