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For how many integers $x$ is the number $x^{4}-51 x^{2}+50$ negative? (  2014 AMC 10B Problems, Question \#20) 
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First, note that $50+1=51$, which motivates us to factor the polynomial as $\left(x^{2}-50\right)\left(x^{2}-1\right)$. Since this expression is negative, one term must be negative and the other positive. Also, the first term is obviously smaller than the second, so $x^{2}-50\textless0\textless x^{2}-1$. Solving this inequality, we find $1\textless x^{2}\textless50$. There are exactly $12$ integers $x$ that satisfy this inequality, $\pm\textbackslash{2,3,4,5,6,7\textbackslash}$. Thus our answer is $(\mathbf{C}) 12$. 