Let's simplify the expression step by step:
Given:
[tex]\[
\left(-v^2 + 5v - 6\right) + \left(v^2 + 7v\right)
\][/tex]
Step 1: Combine like terms
First, let's group the terms involving [tex]\( v^2 \)[/tex], [tex]\( v \)[/tex], and the constant terms separately:
- Terms involving [tex]\( v^2 \)[/tex]:
[tex]\[
-v^2 + v^2
\][/tex]
- Terms involving [tex]\( v \)[/tex]:
[tex]\[
5v + 7v
\][/tex]
- Constant terms:
[tex]\[
-6
\][/tex]
Step 2: Simplify each group
Now, let's simplify each group of terms:
- The [tex]\( v^2 \)[/tex] terms:
[tex]\[
-v^2 + v^2 = 0
\][/tex]
- The [tex]\( v \)[/tex] terms:
[tex]\[
5v + 7v = 12v
\][/tex]
- The constant term remains the same:
[tex]\[
-6
\][/tex]
Step 3: Combine the simplified terms
Combining the simplified terms, we get:
[tex]\[
0 + 12v -6
\][/tex]
Since [tex]\( 0 \)[/tex] does not affect the expression, the simplified expression becomes:
[tex]\[
12v - 6
\][/tex]
Therefore, the simplified expression is:
[tex]\[
\boxed{12v - 6}
\][/tex]
