Simplify using appropriate identities and choose the correct option:

${(\mathrm{a}+\mathrm{b})}^2-{(\mathrm{a}-\mathrm{b})}^2=?$

We will be using these two identities to simplify the given expression:

${(\mathrm{a}+\mathrm{b})}^2={\mathrm{a}}^2+{\mathrm{b}}^2+2\mathrm{ab}$ and ${(\mathrm{a}-\mathrm{b})}^2={\mathrm{a}}^2+{\mathrm{b}}^2-2\mathrm{ab}$

Now, ${(\mathrm{a}+\mathrm{b})}^2-{(\mathrm{a}-\mathrm{b})}^2={\mathrm{a}}^2+{\mathrm{b}}^2+2\mathrm{ab}-({\mathrm{a}}^2+{\mathrm{b}}^2-2\mathrm{ab})$

                                            $={\mathrm{a}}^2+{\mathrm{b}}^2+2\mathrm{ab}-{\mathrm{a}}^2-{\mathrm{b}}^2+2\mathrm{ab}$

                                            $=4\mathrm{ab}$

Hence, ${(\mathrm{a}+\mathrm{b})}^2-{(\mathrm{a}-\mathrm{b})}^2=4\mathrm{ab}$.