Symmetric Matrices & Quadratic Forms · Superchallenge problem

Find the maximum of $2(x_1 + x_2 + x_3 + x_4)(x_1 + 2x_2 + 3x_3 + 4x_4)$ subject to $x_1^2 + x_2^2 + x_3^2 + x_4^2 = 1$.