### Theory:

Polynomial (poly - 'many', nomial - 'term').

When an algebraic expression has one or many terms, then that expression is called a polynomial expression.
In some cases, polynomial multiplication can be performed easier when using the abridged multiplication formulas.
You need to remember $$3$$ formulas:
1) The square of the sum of two expressions:
${\left(a+b\right)}^{2}={a}^{2}+2\mathit{ab}+{b}^{2}$
The square of the sum of two expressions is equal to the square of the first expression, plus twice the product of the first and the second expression, plus the square of the second expression:

$\begin{array}{l}{\left(a+b\right)}^{2}=\left(a+b\right)\cdot \left(a+b\right)=\\ =a\cdot a+a\cdot b+b\cdot a+b\cdot b=\\ ={a}^{2}+\mathit{ab}+\mathit{ba}+{b}^{2}=\\ ={a}^{2}+2\mathit{ab}+{b}^{2}\end{array}$
2) The square of the difference of two expressions:
${\left(a-b\right)}^{2}={a}^{2}-2\mathit{ab}+{b}^{2}$
The square of the difference of two expressions is equal to the square of the first expression, minus twice the product of the first and the second expression, plus the square of the second expression:

$\begin{array}{l}{\left(a-b\right)}^{2}=\left(a-b\right)\cdot \left(a-b\right)=\\ =a\cdot a+a\cdot \left(-b\right)-b\cdot a-b\cdot \left(-b\right)=\\ ={a}^{2}-\mathit{ab}-\mathit{ba}+{b}^{2}=\\ ={a}^{2}-2\mathit{ab}+{b}^{2}\end{array}$
3)  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:

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

${\left(x+3\right)}^{2}={x}^{2}+2\cdot x\cdot 3+{3}^{2}={x}^{2}+6x+9$

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

$\begin{array}{l}{\left(x+3\right)}^{2}=\left(x+3\right)\cdot \left(x+3\right)=\\ =x\cdot x+x\cdot 3+3\cdot x+3\cdot 3=\\ ={x}^{2}+3x+3x+9=\\ ={x}^{2}+6x+9\end{array}$
2)  ${\left(a-b\right)}^{2}={a}^{2}-2\mathit{ab}+{b}^{2}$
Example:
According to the formula:

$\begin{array}{l}{\left(x-3\right)}^{2}=\\ ={x}^{2}-2\cdot x\cdot 3+{3}^{2}=\\ ={x}^{2}-6x+9\end{array}$

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

$\begin{array}{l}{\left(x-3\right)}^{2}=\left(x-3\right)\cdot \left(x-3\right)=\\ =x\cdot x+x\cdot \left(-3\right)-3\cdot x-3\cdot \left(-3\right)=\\ ={x}^{2}-3x-3x+9=\\ ={x}^{2}-6x+9\end{array}$
3)  $\left(a-b\right)\left(a+b\right)={a}^{2}-{b}^{2}$
Example:
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 a formula (multiplying a polynomial by a polynomial):

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

Using the formula, you can get the result much faster.

Important!
The first and the second formula differ only in signs.

${\left(a±b\right)}^{2}={a}^{2}±2\mathit{ab}+{b}^{2}$