Question : The factors of $ x^4+x^2+25$ are:
Option 1: $\left(x^2+3 x-5\right)\left(x^2-3 x+5\right)$
Option 2: $\left(x^2+3 x+5\right)\left(x^2-3 x+5\right)$
Option 3: $\left(x^2-3 x+5\right)\left(x^2-3 x+5\right)$
Option 4: $\left(x^2+3 x+5\right)\left(x^2+3 x+5\right)$
Correct Answer: $\left(x^2+3 x+5\right)\left(x^2-3 x+5\right)$
Solution :
$ x^4+x^2+25$
$= x^4+10x^2-9x^2+25$
$= x^4+10x^2+25-9x^2$
$=(x^2+5)^2-9x^2$
$=(x^2+5)^2-(3x)^2$
$=\left(x^2+3 x+5\right)\left(x^2-3 x+5\right)$
Hence, the answer is $\left(x^2+3 x+5\right)\left(x^2-3 x+5\right)$.
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