Factorize each of the following algebraic expression:

$4 x^{4}+1$

$=4 \mathrm{x}^{4}+4 \mathrm{x}^{2}+1-4 \mathrm{x}^{2}$

$=\left[\left(2 \mathrm{x}^{2}\right)^{2}+2 \times 2 \mathrm{x}^{2} \times 1+1\right]-4 \mathrm{x}^{2}$

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

$=\left[\left(2 \mathrm{x}^{2}+1\right)-2 \mathrm{x}\right]\left[\left(2 \mathrm{x}^{2}+1\right)+2 \mathrm{x}\right]$

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