The given expression is:

\[
\frac{7v^2 + 3v}{-v}
\]

We can simplify this step by step.

1. **Factor the numerator**: Notice that \(v\) is common in both terms in the numerator.

\[
7v^2 + 3v = v(7v + 3)
\]

2. **Simplify the expression** by dividing the entire numerator by \(-v\):

\[
\frac{v(7v + 3)}{-v} = -(7v + 3)
\]

Thus, the simplified expression is:

\[
-(7v + 3) = -7v - 3
\]

### Final answer:
\[
-7v - 3
\]

Would you like more details or explanations? Here are 5 related questions:

1. How would this change if the expression were \(\frac{7v^2 + 3v}{v}\)?
2. Can you apply similar steps to simplify \(\frac{6x^2 - 4x}{2x}\)?
3. What if the denominator were \(-v^2\)? How would that affect the simplification?
4. How can this simplification be applied in solving quadratic equations?
5. How would this expression change if \(v\) were a negative number?

**Tip**: Always check for common factors in the numerator when simplifying expressions!

Simplify the rational expression (7v^2 + 3v) / (-v) with a step-by-step guide. This solution explains factoring, common factors, and simplifying rational expressions, suitable for Grades 8-10 students learning algebra.

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