Which of the following is the factors of ${\mathrm{x}}^2-{\mathrm{y}}^2-2\mathrm{y}-1$?

Given, ${\mathrm{x}}^2-{\mathrm{y}}^2-2\mathrm{y}-1$ 

Simplifying the given expression, we get

$\Rightarrow {\mathrm{x}}^2-{\mathrm{y}}^2-2\mathrm{y}-1$

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

$\Rightarrow {\mathrm{x}}^2-\left({\left(\mathrm{y}\right)}^2+2\times \mathrm{y}\times 1+{\left(1\right)}^2\right)$

Using the identity ${\mathbf{a}}^{\mathbf{2}}\mathbf{+}\mathbf{2}\mathbf{ab}\mathbf{+}{\mathbf{b}}^{\mathbf{2}}\mathbf{=}{\left(a+b\right)}^{\mathbf{2}}$

$\Rightarrow {\left(\mathrm{x}\right)}^2-{\left(\mathrm{y}+1\right)}^2$
 Using the identity ${\mathbf{a}}^{\mathbf{2}}\mathbf{-}{\mathbf{b}}^{\mathbf{2}}\mathbf{=}\left(a+b\right)\left(a-b\right)$

$\Rightarrow \left(\mathrm{x}-\left(\mathrm{y}+1\right)\right)\left(\mathrm{x}+\left(\mathrm{y}+1\right)\right)$

$\Rightarrow \left(\mathrm{x}-\mathrm{y}-1\right)\left(\mathrm{x}+\mathrm{y}+1\right)$ 

Hence, factors of ${\mathbf{x}}^{\mathbf{2}}\mathbf{-}{\mathbf{y}}^{\mathbf{2}}\mathbf{-}\mathbf{2}\mathbf{y}\mathbf{-}\mathbf{1}\mathbf{ }\mathbf{are}\mathbf{ }\mathbf{(}\mathbf{x}\mathbf{-}\mathbf{y}\mathbf{-}\mathbf{1}\mathbf{)}\mathbf{(}\mathbf{x}\mathbf{+}\mathbf{y}\mathbf{+}\mathbf{1}\mathbf{)}$