Question

$x {}^{2} \times (y {}^{2} - z {}^{2} ) + y {}^{2} (z {}^{2} - y {}^{2} ) + z {}^{2} (x {}^{2} - y {}^{2} )$
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Answer 1

If you mean the question is this:

$x^{2} \times (y^{2} - z^{2} ) + y^{2} \times (z^{2} - x^{2}) + z^{2} \times (x^{2} - y^{2})$

then the answer's this:

$= x^{2}y^{2} - x^{2} z^{2} + y^{2}z^{2} - x^{2} y^{2} + x^{2} z^{2} - y^{2}z^{2}$

$= x^{2}y^{2} - x^{2} y^{2} + y^{2}z^{2} - y^{2} z^{2} + x^{2} z^{2} - x^{2}z^{2}$

$= 0$

If you mean the question is this:

$x^{2} \times (y^{2} - z^{2} ) + y^{2} \times (z^{2} - y^{2}) + z^{2} \times (x^{2} - y^{2})$

then the answer's this:

$= x^{2}y^{2} - x^{2} z^{2} + y^{2}z^{2} - y^{4} + x^{2} z^{2} - y^{2}z^{2}$

$= x^{2}y^{2} - y^{4} + y^{2}z^{2} - y^{2} z^{2} + x^{2} z^{2} - x^{2}z^{2}$

$= x^{2}y^{2} - y^{4}$