Factorize each of the following expression:

$(2 \mathrm{x}+1)^{2}-9 \mathrm{x}^{4}$

$=(2 \mathrm{x}+1)^{2}-\left(3 \mathrm{x}^{2}\right)^{2}$

$=\left[(2 \mathrm{x}+1)-3 \mathrm{x}^{2}\right]\left[(2 \mathrm{x}+1)+3 \mathrm{x}^{2}\right]$

$=\left(-3 \mathrm{x}^{2}+2 \mathrm{x}+1\right)\left(3 \mathrm{x}^{2}+2 \mathrm{x}+1\right)$

We can factorise the quadratic expressions in the curved brackets as:

$\left(-3 \mathrm{x}^{2}+3 \mathrm{x}-\mathrm{x}+1\right)\left(3 \mathrm{x}^{2}+2 \mathrm{x}+1\right)$

$=\{3 \mathrm{x}(-\mathrm{x}+1)+1(-\mathrm{x}+1)\}\left(3 \mathrm{x}^{2}+2 \mathrm{x}+1\right)$

$=(-\mathrm{x}+1)(3 \mathrm{x}+1)\left(3 \mathrm{x}^{2}+2 \mathrm{x}+1\right)$

$=-(\mathrm{x}-1)(3 \mathrm{x}+1)\left(3 \mathrm{x}^{2}+2 \mathrm{x}+1\right)$