Show that

$(\vec{a}-\vec{b}) \times(\vec{a}+\vec{b})$

$=(\vec{a}-\vec{b}) \times \vec{a}+(\vec{a}-\vec{b}) \times \vec{b} \quad$ [By distributivity of vector product over addition]

$=\vec{a} \times \vec{a}-\vec{b} \times \vec{a}+\vec{a} \times \vec{b}-\vec{b} \times \vec{b} \quad$ [Again, by distributivity of vector product over addition]

$=\overrightarrow{0}+\vec{a} \times \vec{b}+\vec{a} \times \vec{b}-\overrightarrow{0}$

$=2 \vec{a} \times \vec{b}$