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Simplify each of the following products:
\( (x^{2}+x-2)(x^{2}-x+2) \)
Given:
\( (x^{2}+x-2)(x^{2}-x+2) \)
To do:
We have to simplify the given product.
Solution:
We know that,
$(a+b)^2=a^2+b^2+2ab$
$(a-b)^2=a^2+b^2-2ab$
$(a+b)(a-b)=a^2-b^2$
Therefore,
$(x^{2}+x-2)(x^{2}-x+2)=[x^{2}+(x-2)][x^{2}-(x-2)]$
$=(x^{2})^{2}-(x-2)^{2}$
$=x^{4}-[(x)^{2}-2 \times x \times 2+(2)^{2}]$
$=x^{4}-(x^{2}-4 x+4)$
$=x^{4}-x^{2}+4 x-4$
Hence, $(x^{2}+x-2)(x^{2}-x+2)=x^{4}-x^{2}+4 x-4$.
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