Let's solve the equation step by step.

Given:

\[
(2x + 1)(3x - 7) = (2x - 2)(3x + 8)
\]

### Step 1: Expand both sides of the equation

First, expand the left side:

\[
(2x + 1)(3x - 7) = 2x \cdot 3x + 2x \cdot (-7) + 1 \cdot 3x + 1 \cdot (-7)
\]

\[
= 6x^2 - 14x + 3x - 7
\]

\[
= 6x^2 - 11x - 7
\]

Now, expand the right side:

\[
(2x - 2)(3x + 8) = 2x \cdot 3x + 2x \cdot 8 - 2 \cdot 3x - 2 \cdot 8
\]

\[
= 6x^2 + 16x - 6x - 16
\]

\[
= 6x^2 + 10x - 16
\]

So the equation becomes:

\[
6x^2 - 11x - 7 = 6x^2 + 10x - 16
\]

### Step 2: Simplify the equation

Subtract \(6x^2\) from both sides:

\[
-11x - 7 = 10x - 16
\]

### Step 3: Combine like terms

Add \(11x\) to both sides:

\[
-7 = 21x - 16
\]

Add \(16\) to both sides:

\[
9 = 21x
\]

### Step 4: Solve for \(x\)

Divide both sides by 21:

\[
x = \frac{9}{21} = \frac{3}{7}
\]

### Final Answer:
\[
x = \frac{3}{7}
\]

Would you like further details on the solution, or do you have any questions?

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Here are 5 related questions for you to consider:

1. What happens if the original equation had different coefficients?
2. Can you solve a similar equation where the right-hand side is a quadratic trinomial?
3. How would the solution process change if the equation were not factorable?
4. Can you solve this equation by graphing both sides and finding the intersection point?
5. How would you solve a similar problem if it were set up as an inequality instead of an equation?

**Tip:** Always check your solution by substituting the value of \(x\) back into the original equation to verify that both sides are equal.

Learn how to solve the equation (2x + 1)(3x - 7) = (2x - 2)(3x + 8) step-by-step with detailed explanations and examples. This problem involves factoring quadratic expressions and applying algebraic methods. Suitable for high school students.

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