Polar form of a quadratic form

Notation. $u=(x_1,y_1)$, $v=(x_2,y_2)$.

Compute.
$$Q(u+v)=(x_1+x_2)^2+4(x_1+x_2)(y_1+y_2)+3(y_1+y_2)^2.$$
Expand:
$$=x_1^2+2x_1x_2+x_2^2+4x_1y_1+4x_1y_2+4x_2y_1+4x_2y_2+3y_1^2+6y_1y_2+3y_2^2.$$

Subtract $Q(u)+Q(v)$. The pure terms cancel:
$$Q(u+v)-Q(u)-Q(v)=2x_1x_2+4x_1y_2+4x_2y_1+6y_1y_2.$$

Divide by 2.
$$\varphi(u,v)=x_1x_2+2x_1y_2+2x_2y_1+3y_1y_2.$$

Check. $\varphi(u,u)=x_1^2+4x_1y_1+3y_1^2=Q(u)$. Correct.

Conclusion. The polar form is $\varphi(u,v)=x_1x_2+2x_1y_2+2x_2y_1+3y_1y_2$.