#### Answer

$=\left( \cot ^{2}x+1\right) \left( \cot ^{2}x+2\right) $

#### Work Step by Step

$\cot ^{4}x+3\cot ^{2}x+2=\cot ^{4}x+2\cot ^{2}x+1+\left( 1+\cot ^{2}x\right) =\left( \cot ^{2}x+1\right) ^{2}+\left( \cot ^{2}x+1\right) =\left( \cot ^{2}x+1\right) \left( \cot ^{2}x+1+1\right) $
$=\left( \cot ^{2}x+1\right) \left( \cot ^{2}x+2\right) $