To find the term that completes the product \((-5x - 3)(-5x + \_\_\_\_\_) \) so that it becomes a difference of squares, let's consider the nature of difference of squares.
The difference of squares formula is given by:
[tex]\[ a^2 - b^2 = (a - b)(a + b) \][/tex]
Here, we want our expression to take the form of \( (-5x - 3)(-5x + \text{some term}) \). For it to be a difference of squares, the second binomial should be in the form \( (-5x + b) \), and our end product should look like:
[tex]\[ (-5x)^2 - (\text{some term})^2 \][/tex]
Now let's break down the terms.
1. \( \text{First term squared} \):
[tex]\[
(-5x) \times (-5x) = 25x^2
\][/tex]
Thus, the first term squared is 25x^2.
2. \( \text{Second term squared} \):
We know from analyzing the numbers and the result given:
[tex]\[
\text{Second term's contribution is } -9
\][/tex]
So, we need to figure out what squared term will give the numeric result that matches.
To find our value \( b \):
[tex]\[
b^2 = \left(3\right)^2 = 9
\][/tex]
Since we are ensuring the second term completes a difference of squares, the numerical part added to the product should be appropriately balanced such that it resembles the \((-5 x - 3)\) counterpart forming.
Now, combining all findings, the term \( b \) can be:
[tex]\[
b = \pm 3 \text{ or } \pm 9
\][/tex]
We validate this as:
[tex]\[
(-5 x - 3)( - 5 x + [3 \text{ or } -3] \text{ or } [9 \text{ or } -9])
\][/tex]
From the choices, only one number will fit properly into the product ensuring \( b \):
[tex]\[
a^2 - b^2 = 25x^2 - 9
\][/tex]
Finally, this elaboration shows:
[tex]\[
\boxed{9}
\][/tex]
is the term that completes the product so it is the difference of squares.
