To find the missing coefficient of the [tex]\( x \)[/tex]-term in the product [tex]\( (-x - 5)^2 \)[/tex] after it has been simplified, let's proceed step by step:
1. Write the expression:
[tex]\[
(-x - 5)^2
\][/tex]
2. Expand the expression:
We use the distributive property (also known as the FOIL method for binomials):
[tex]\[
(-x - 5)^2 = (-x - 5)(-x - 5)
\][/tex]
3. Apply the FOIL method:
- First term: Multiply the first terms in each binomial:
[tex]\[
(-x) \cdot (-x) = x^2
\][/tex]
- Outer term: Multiply the outer terms in the product:
[tex]\[
(-x) \cdot (-5) = 5x
\][/tex]
- Inner term: Multiply the inner terms in the product:
[tex]\[
(-5) \cdot (-x) = 5x
\][/tex]
- Last term: Multiply the last terms in each binomial:
[tex]\[
(-5) \cdot (-5) = 25
\][/tex]
4. Combine all the terms:
[tex]\[
x^2 + 5x + 5x + 25
\][/tex]
5. Combine like terms:
- Combine the [tex]\( x \)[/tex]-terms:
[tex]\[
5x + 5x = 10x
\][/tex]
- Therefore, the expanded expression is:
[tex]\[
x^2 + 10x + 25
\][/tex]
6. Identify the coefficient of the [tex]\( x \)[/tex]-term:
The coefficient of the [tex]\( x \)[/tex]-term in the simplified expression [tex]\( x^2 + 10x + 25 \)[/tex] is [tex]\( 10 \)[/tex].
Therefore, the missing coefficient of the [tex]\( x \)[/tex]-term in the product [tex]\( (-x - 5)^2 \)[/tex] after it has been simplified is [tex]\( \boxed{10} \)[/tex].
