Solve the equation 5x - ||v|| v = ||w||(w-5x) for x with v = (1, 2, -4, 0) and w = (-3, 5, 1, 1 )
so you have $\displaystyle ||v|| = \sqrt{1^2 + 2^2 + 4^2 + 0^2} = \sqrt{21} $ and $\displaystyle ||w|| = \sqrt{9 + 25 + 1 + 1} = \sqrt{36} = 6 $
$\displaystyle 5x - \sqrt{21}v = 6w - 30x $ [
$\displaystyle 35x = 6w + \sqrt{21} v $
right side equals $\displaystyle \begin{bmatrix}-18 + \sqrt{21} \\ 30 + 2 * \sqrt{21} \\ 6 - 4*\sqrt{21} \\ 6 \end{bmatrix} $
multiply the right side by $\displaystyle \frac{1}{35} $
and that is your x.
x = $\displaystyle \frac{1}{35} \begin{bmatrix}-18 + \sqrt{21} \\ 30 + 2 * \sqrt{21} \\ 6 - 4*\sqrt{21} \\ 6 \end{bmatrix} $