To solve the expression [tex]\(5(a+b)+2(a+c)-4(b+c)\)[/tex], we will follow a step-by-step approach to simplify it.
### Step-by-Step Simplification:
1. Expand each term:
The expression begins as:
[tex]\[
5(a + b) + 2(a + c) - 4(b + c)
\][/tex]
We will first distribute the coefficients inside the parentheses:
[tex]\[
5(a + b) = 5a + 5b
\][/tex]
[tex]\[
2(a + c) = 2a + 2c
\][/tex]
[tex]\[
-4(b + c) = -4b - 4c
\][/tex]
2. Combine all the expanded terms:
Now, substitute the expanded terms back into the original expression:
[tex]\[
5a + 5b + 2a + 2c - 4b - 4c
\][/tex]
3. Group like terms together:
Combine all terms involving [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] separately:
[tex]\[
(5a + 2a) + (5b - 4b) + (2c - 4c)
\][/tex]
Simplifying each group:
[tex]\[
5a + 2a = 7a
\][/tex]
[tex]\[
5b - 4b = b
\][/tex]
[tex]\[
2c - 4c = -2c
\][/tex]
4. Write the simplified expression:
Therefore, we get:
[tex]\[
7a + b - 2c
\][/tex]
### Final Answer:
The simplified form of the expression [tex]\(5(a+b)+2(a+c)-4(b+c)\)[/tex] is:
[tex]\[
7a + b - 2c
\][/tex]
So the correct choice from the given options is [tex]\( \boxed{7a + b - 2c} \)[/tex].
