We are given the equation:

\[
3(q - 2) + q^2 = q(q + 3) - 6
\]

### Step 1: Expand both sides
Expand the left-hand side:
\[
3(q - 2) + q^2 = 3q - 6 + q^2
\]

Expand the right-hand side:
\[
q(q + 3) - 6 = q^2 + 3q - 6
\]

So the equation becomes:
\[
3q - 6 + q^2 = q^2 + 3q - 6
\]

### Step 2: Simplify both sides
Cancel out the \(q^2\) terms and the constants:
\[
3q - 6 = 3q - 6
\]

Since both sides are the same, this equation is **always true** for any value of \(q\).

### Conclusion:
The solution to this equation is that **any value of \(q\) is a solution**, meaning there are **infinite solutions**.

Would you like more details or have any questions?

### Relative Questions:
1. What happens when solving a quadratic equation with no real solutions?
2. How can you identify when an equation has infinite solutions?
3. Can this method be applied to other types of equations?
4. How do you verify the solution of an equation like this?
5. What are the implications of variables canceling out in algebra?

**Tip**: Always simplify equations step by step to avoid errors when solving.

This page explains how to solve the equation 3(q - 2) + q^2 = q(q + 3) - 6, showcasing algebraic concepts such as the distributive property and quadratic expansion. The solution demonstrates how to identify infinite solutions. Ideal for students in Grades 9-11.

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