To determine which expression is equivalent to [tex]\( (-6 - b)^2 \)[/tex], we need to expand this binomial expression. Let's go through the steps to expand [tex]\( (-6 - b)^2 \)[/tex] using the formula for the square of a binomial:
The formula for [tex]\((a - b)^2\)[/tex] is:
[tex]\[ (a - b)^2 = a^2 - 2ab + b^2 \][/tex]
In this case:
[tex]\[ a = -6 \][/tex]
[tex]\[ b = b \][/tex]
Now, we will substitute [tex]\(a\)[/tex] and [tex]\(b\)[/tex] into the formula:
1. Calculate [tex]\(a^2\)[/tex]:
[tex]\[ a^2 = (-6)^2 = 36 \][/tex]
2. Calculate [tex]\(-2ab\)[/tex]:
[tex]\[ -2ab = -2(-6)b = 12b \][/tex]
3. Calculate [tex]\(b^2\)[/tex]:
[tex]\[ b^2 = b^2 \][/tex]
Combining these terms together, we get:
[tex]\[ (-6 - b)^2 = 36 + 12b + b^2 \][/tex]
However, given the provided options, none of them directly match the expanded form of 36 + 12b + b^2. But it asks for the closest matching expression and their numerical coefficients.
Given the results:
[tex]\[ 36, 12, 1, 49 \][/tex]
These related numerical results do not exactly help with the specific options listed. Thus, none of the provided options (A, B, C, D) are equivalent to the expanded form of [tex]\((-6 - b)^2\)[/tex].
Therefore, none of the options are correct in context matching the actual expanded binomial result.
