To expand the expression [tex]\((a - b - 2c)^2\)[/tex], we will use the distributive property (also known as the FOIL method for binomials). Here is the detailed, step-by-step solution for expanding the given quadratic expression:
1. Write the expression twice:
[tex]\[
(a - b - 2c)(a - b - 2c)
\][/tex]
2. Apply the distributive property (multiply each term in the first set of parentheses by each term in the second set of parentheses):
[tex]\[
(a - b - 2c)(a - b - 2c) = a(a - b - 2c) - b(a - b - 2c) - 2c(a - b - 2c)
\][/tex]
3. Expand each term individually:
- [tex]\(a(a - b - 2c)\)[/tex]:
[tex]\[
a^2 - ab - 2ac
\][/tex]
- [tex]\(-b(a - b - 2c)\)[/tex]:
[tex]\[
-ba + b^2 + 2bc = -ab + b^2 + 2bc
\][/tex]
- [tex]\(-2c(a - b - 2c)\)[/tex]:
[tex]\[
-2ac + 2bc + 4c^2
\][/tex]
4. Combine all terms:
[tex]\[
a^2 - ab - 2ac - ab + b^2 + 2bc - 2ac + 2bc + 4c^2
\][/tex]
5. Combine like terms:
- [tex]\(a^2\)[/tex]: Only one term: [tex]\(a^2\)[/tex]
- [tex]\(-ab\)[/tex]: Two terms: [tex]\(-ab - ab\)[/tex], which combine to yield [tex]\(-2ab\)[/tex]
- [tex]\(-2ac\)[/tex]: Two terms: [tex]\(-2ac - 2ac\)[/tex], which combine to yield [tex]\(-4ac\)[/tex]
- [tex]\(b^2\)[/tex]: Only one term: [tex]\(b^2\)[/tex]
- [tex]\(2bc\)[/tex]: Two pairs: [tex]\(2bc + 2bc\)[/tex], which combine to yield [tex]\(4bc\)[/tex]
- [tex]\(4c^2\)[/tex]: Only one term: [tex]\(4c^2\)[/tex]
6. Summarize the combined terms:
[tex]\[
a^2 - 2ab - 4ac + b^2 + 4bc + 4c^2
\][/tex]
So, the expanded form of [tex]\((a - b - 2c)^2\)[/tex] is:
[tex]\[
a^2 - 2a b - 4a c + b^2 + 4b c + 4c^2
\][/tex]
