Find a change in variable that will reduce the quadratic form to a sum of squares Find a change of variable that will reduce the quadratic form
$x_1^2-x_3^2-4x_1x_2+4x_2x_3$
to a sum of squares, and express the quadratic form in terms of the new variable.

$$
x^2-z^2-4xy-4yz = x^2 +4y^2 -4xy - \left(4y^2+z^2+4yz\right)
$$
and now you can trivially complete the square. Now substitute to get the sum of squares.
Note it's not expected to be the same sum of squares as @Gerry Myerson's answer.
