### Theory:

The difference of squares of two expressions:
$\left(a-b\right)\left(a+b\right)={a}^{2}-{b}^{2}$
The product of the sum and difference of two expressions is equal to the difference of the squares of these expressions:

$\begin{array}{l}\left(a-b\right)\cdot \left(a+b\right)=\\ =a\cdot a+a\cdot b-b\cdot a-b\cdot b=\\ ={a}^{2}\overline{)+\mathit{ab}}\overline{)-\mathit{ab}}-{b}^{2}=\\ ={a}^{2}-{b}^{2}\end{array}$
Application of the formula $\left(a-b\right)\left(a+b\right)={a}^{2}-{b}^{2}$
Example:
1) According to the formula:

$\begin{array}{l}\left(x-3\right)\left(x+3\right)=\\ ={x}^{2}-{3}^{2}=\\ ={x}^{2}-9\end{array}$

Without the formula (multiplying a polynomial by a polynomial):

$\begin{array}{l}\left(x-3\right)\left(x+3\right)=\\ =x\cdot x+x\cdot 3-3\cdot x-3\cdot 3=\\ ={x}^{2}\overline{)+3x}\overline{)-3x}-9=\\ ={x}^{2}-9\end{array}$

2) According to the formula:

$\begin{array}{l}\left(4x-y\right)\left(4x+y\right)=\\ ={\left(4x\right)}^{2}-{y}^{2}=\\ =16{x}^{2}-{y}^{2}\end{array}$

Without the formula (multiplying a polynomial by a polynomial):

$\begin{array}{l}\left(4x-y\right)\left(4x+y\right)=\\ =4x\cdot 4x+4x\cdot y-y\cdot 4x-y\cdot y=\\ =16{x}^{2}\overline{)+4\mathit{xy}}\overline{)-4\mathit{xy}}-{y}^{2}=\\ =16{x}^{2}-{y}^{2}\end{array}$

3) According to the formula:

$\begin{array}{l}\left(6z-9\right)\left(6z+9\right)=\\ ={\left(6z\right)}^{2}-{9}^{2}=\\ ={36z}^{2}-81\end{array}$

Without the formula (multiplying a polynomial by a polynomial):

$\begin{array}{l}\left(6z-9\right)\left(6z+9\right)=\\ =6z\cdot 6z+6z\cdot 9-9\cdot 6z-9\cdot 9=\\ =36{z}^{2}\overline{)+54z}\overline{)-54z}-81=\\ =36{z}^{2}-81\end{array}$