### Theory:

Identity $$- 4$$:

Let us first, simplify the identity .

Multiply the expression, as shown below.

Now, we construct a figure to understand the concept.

Then we construct a rectangle using the above information.

In the given figure, $$AB = AD = a$$.

So, the area of square $$ABCD = a^2$$.

Also, $$SB = DP = b$$. Then the area of the rectangle $$SBCT = ab$$.

Similarly, the area of the rectangle $$DPRC = ab$$. And the area of the square $$TQRC = b^2$$.

Area of the rectangle $$DPQT = ab − b^2$$.

Hence, $$\text{the area of the rectangle APQS = The area of square ABCD}$$ $$\text{– area of rectangle STCB}$$ $$\text{+ area of rectangle DPQT}$$.

$\begin{array}{l}={a}^{2}-\mathit{ab}+\left(\mathit{ab}-{b}^{2}\right)\\ \\ ={a}^{2}-\overline{)\mathit{ab}}+\overline{)\mathit{ab}}-{b}^{2}\\ \\ ={a}^{2}-{b}^{2}\end{array}$

Therefore, .
Example:
Simplify  using the identity.

First, develop the given  expression using the identity .

Here, $$a = 3x$$; $$b = 9y$$.

Therefore,  $$=$$ 9$$x^2 -$$81.