To determine the missing factor, let's start by recognizing the form of the given binomial:
[tex]\[
x^2 - 49
\][/tex]
The expression [tex]\( x^2 - 49 \)[/tex] fits the pattern of a difference of squares, which is a type of algebraic expression that can be factored as follows:
[tex]\[
a^2 - b^2 = (a + b)(a - b)
\][/tex]
For our given binomial [tex]\( x^2 - 49 \)[/tex], we can identify [tex]\( a \)[/tex] and [tex]\( b \)[/tex] as follows:
[tex]\[
a = x \quad \text{and} \quad b = 7 \quad \text{because} \quad 49 = 7^2
\][/tex]
Using the formula for the difference of squares, we can factor [tex]\( x^2 - 49 \)[/tex]:
[tex]\[
x^2 - 49 = (x + 7)(x - 7)
\][/tex]
Given that one of the factors is [tex]\( (x + 7) \)[/tex], the missing factor must be:
[tex]\[
(x - 7)
\][/tex]
To verify, we can multiply the factors together to check that they yield the original binomial:
[tex]\[
(x + 7)(x - 7) = x^2 - 7x + 7x - 49 = x^2 - 49
\][/tex]
Since this multiplication returns the original binomial [tex]\( x^2 - 49 \)[/tex], our factorization is correct.
Therefore, the missing factor is:
[tex]\[
\boxed{(x - 7)}
\][/tex]
