Add the following
$x^{2} + y^{2} - z^{2}$ and ${- x}^{2} - y^{2} + z^{2}$

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Solution

$x^{2} + y^{2} - z^{2} + (- x^{2} - y^{2} + z^{2})$
$= x^{2} + y^{2} - z^{2} - x^{2} - y^{2} + z^{2}$
$= 0$ since like terms get cancelled.

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