Given the equation [tex]\( a + b + c = 0 \)[/tex], we aim to find the value of [tex]\( (a + b)(b + d(c + 1)) \)[/tex].
### Step-by-Step Solution:
1. Substituting values:
Let’s assume some values:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = 2 \)[/tex]
2. Determining [tex]\( c \)[/tex]:
Since [tex]\( a + b + c = 0 \)[/tex], substituting the values we get:
[tex]\[
1 + 2 + c = 0
\][/tex]
Solving for [tex]\( c \)[/tex], we have:
[tex]\[
3 + c = 0
\][/tex]
[tex]\[
c = -3
\][/tex]
3. Assuming another variable [tex]\( d \)[/tex]:
Let [tex]\( d = 3 \)[/tex].
4. Evaluate [tex]\( a + b \)[/tex]:
[tex]\[
a + b = 1 + 2 = 3
\][/tex]
5. Evaluate [tex]\( c + 1 \)[/tex]:
[tex]\[
c + 1 = -3 + 1 = -2
\][/tex]
6. Evaluate [tex]\( d(c + 1) \)[/tex]:
[tex]\[
d(c + 1) = 3 \times (-2) = -6
\][/tex]
7. Evaluate [tex]\( b + d(c + 1) \)[/tex]:
[tex]\[
b + d(c + 1) = 2 + (-6) = 2 - 6 = -4
\][/tex]
8. Finally, calculate [tex]\( (a + b)(b + d(c + 1)) \)[/tex]:
[tex]\[
(a + b)(b + d(c + 1)) = 3 \times (-4) = -12
\][/tex]
### Recap of final values:
- [tex]\( a + b = 3 \)[/tex]
- [tex]\( b + d(c + 1) = -4 \)[/tex]
- The final result is:
[tex]\[
(a + b)(b + d(c + 1)) = -12
\][/tex]
Thus, the value of [tex]\( (a + b)(b + d(c + 1)) \)[/tex] is [tex]\(-12\)[/tex].
