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**Hint:**In order to determine the product of the $\left( {b + 7} \right)\left( {b - 7} \right)$, we will use the difference of the squares of the two variables from the algebraic identities. Then compare both the given and the formula to determine the value of the variables and substitute in the formula to determine the required.

**Complete step-by-step solution:**

Here, we need to find the product of $\left( {b + 7} \right)\left( {b - 7} \right)$.

We know from the formula of difference of squares of two variables is,

${x^2} - {y^2} = \left( {x + y} \right)\left( {x - y} \right)$

Now, comparing the given with the formula, we have,

$x = b$ and $y = 7$

Substituting the values in the formula we have,

${b^2} - {7^2} = \left( {b + 7} \right)\left( {b - 7} \right)$

Hence, $\left( {b + 7} \right)\left( {b - 7} \right) = {b^2} - 49$

**Note:**Algebra includes both numbers and letters. Numbers are fixed, i.e. their value is known. Letters or alphabets are used to represent the unknown quantities in the algebra formula. Now, a combination of numbers, letters, factorials, matrices etc. is used to form an equation or formula. This is essentially the methodology for algebra. As students study for their exams, there are certain very important algebra formulas and equations that they must learn. These formulas are the cornerstone of basic or elementary algebra. Only learning the formulas is not sufficient. The students must also understand the concept behind the formula and learn to apply them correctly.

Alternatively, we can also use the FOIL method to determine the product of the given. The letters FOIL stands for First, Outer, Inner, Last. First means multiply the terms which occur first in each binomial, Outer means multiply the outermost terms in the product, Inner means multiply the innermost terms, and Last means multiply the terms which occur last in each binomial. By following this procedure, we can solve the given.

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