To simplify the expression [tex]\((a - b - 2c)^2\)[/tex], let's expand it step by step.
1. Write the expression:
[tex]\[
(a - b - 2c)^2
\][/tex]
2. Apply the formula for the square of a binomial:
The formula for the square of a binomial [tex]\((x - y)^2\)[/tex] is:
[tex]\[
(x - y)^2 = x^2 - 2xy + y^2
\][/tex]
In our case, the binomial is [tex]\(a - b - 2c\)[/tex]. Let's take [tex]\(x\)[/tex] as [tex]\(a\)[/tex] and [tex]\(y\)[/tex] as [tex]\(b + 2c\)[/tex].
3. Substitute the values:
[tex]\[
(a - (b + 2c))^2
\][/tex]
4. Expand using the binomial square formula:
[tex]\[
(a - (b + 2c))^2 = a^2 - 2a(b + 2c) + (b + 2c)^2
\][/tex]
5. Distribute and simplify:
[tex]\[
= a^2 - 2a(b + 2c) + (b + 2c)^2
\][/tex]
[tex]\[
= a^2 - 2ab - 4ac + (b + 2c)^2
\][/tex]
6. Expand the second binomial:
[tex]\[
(b + 2c)^2 = b^2 + 4bc + 4c^2
\][/tex]
7. Combine all terms:
[tex]\[
a^2 - 2ab - 4ac + b^2 + 4bc + 4c^2
\][/tex]
8. Write the final expanded expression:
[tex]\[
(a - b - 2c)^2 = a^2 - 2ab - 4ac + b^2 + 4bc + 4c^2
\][/tex]
Thus, the expanded form of [tex]\((a - b - 2c)^2\)[/tex] is:
[tex]\[
a^2 - 2ab - 4ac + b^2 + 4bc + 4c^2
\][/tex]
