Multiply: $- x y^{2} z^{2} + 2 y z^{2} x + 3 x^{2} y z - 7$ by $- 6 x^{2} y z$

+ 6 x^{3} y^{3} z^{3} - 12 x^{3} y^{2} z^{3} - 18 x^{4} y^{2} z^{2} + 42 x^{2} y z

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+ 6 x^{3} y^{3} z^{3} - 12 x^{3} y^{2} z^{3} - 18 x^{4} y^{2} z^{2} - 42 x^{2} y z

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+ 6 x^{3} y^{3} z^{3} + 12 x^{3} y^{2} z^{3} - 18 x^{4} y^{2} z^{2} + 42 x^{2} y z

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None of these

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Solution

The correct option is B $+ 6 x^{3} y^{3} z^{3} - 12 x^{3} y^{2} z^{3} - 18 x^{4} y^{2} z^{2} + 42 x^{2} y z$
To multiply the given polynomials $- x y^{2} z^{2} + 2 y z^{2} x + 3 x^{2} y z - 7$ and $- 6 x^{2} y z$ , we distribute each term of the first polynomial to every term of the second polynomial as shown below:

$- 6 x^{2} y z (- x y^{2} z^{2} + 2 y z^{2} x + 3 x^{2} y z - 7) = (- 6 x^{2} y z \times - x y^{2} z^{2}) + (- 6 x^{2} y z \times 2 y z^{2} x) + (- 6 x^{2} y z \times 3 x^{2} y z) + (- 6 x^{2} y z \times - 7) = 6 x^{3} y^{3} z^{3} - 12 x^{3} y^{2} z^{3} - 18 x^{4} y^{2} z^{2} + 42 x^{2} y z$

Hence, $- 6 x^{2} y z (- x y^{2} z^{2} + 2 y z^{2} x + 3 x^{2} y z - 7) = 6 x^{3} y^{3} z^{3} - 12 x^{3} y^{2} z^{3} - 18 x^{4} y^{2} z^{2} + 42 x^{2} y z$

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