# Factor completely each polynomial, and indicate any that are not

Factor completely each polynomial, and indicate any that are not factorable using integers. ${x}^{4}+6{x}^{2}+9$
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Step 1
The given polynomial is ${x}^{4}+6{x}^{2}+9$
Step 2
Let ${x}^{2}=t$ we have
${x}^{4}+6{x}^{2}+9={t}^{2}+6t+9$
$={t}^{2}+3t+3t+9$
=t(t+3)+3(t+3)
=(t+3)(t+3)
$={\left(t+3\right)}^{2}$
Resubstituting $t={x}^{2}$ we have
${\left(t+3\right)}^{2}={\left({x}^{2}+3\right)}^{2}$
Factors of given equation is
${x}^{4}+6{x}^{2}+9=\left({x}^{2}+3\right)\left({x}^{2}+3\right)$
Solving the equation we get'
${\left({x}^{2}+3\right)}^{2}=0$
$\left({x}^{2}+3\right)=0$
${x}^{2}=-3$
$x=\sqrt{-3}$
Therefore, no real solutions possible for given equation

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