We are given the equation:

\[
\frac{2a - 3}{a - 1} = \frac{2a + 3}{a + 1} + \frac{10}{a^2 - 1}
\]

### Step 1: Factor the denominator \( a^2 - 1 \)
Notice that \( a^2 - 1 \) is a difference of squares, so we can factor it:

\[
a^2 - 1 = (a - 1)(a + 1)
\]

This allows us to rewrite the equation as:

\[
\frac{2a - 3}{a - 1} = \frac{2a + 3}{a + 1} + \frac{10}{(a - 1)(a + 1)}
\]

### Step 2: Find a common denominator
The common denominator for all terms is \( (a - 1)(a + 1) \).

#### Left side:
We need to multiply the left side by \( (a + 1) \):

\[
\frac{2a - 3}{a - 1} \cdot \frac{a + 1}{a + 1} = \frac{(2a - 3)(a + 1)}{(a - 1)(a + 1)}
\]

Expanding the numerator:

\[
(2a - 3)(a + 1) = 2a^2 + 2a - 3a - 3 = 2a^2 - a - 3
\]

So, the left side becomes:

\[
\frac{2a^2 - a - 3}{(a - 1)(a + 1)}
\]

#### Right side:
We already have the common denominator \( (a - 1)(a + 1) \), so we can add the terms:

\[
\frac{2a + 3}{a + 1} \cdot \frac{a - 1}{a - 1} = \frac{(2a + 3)(a - 1)}{(a - 1)(a + 1)}
\]

Expanding the numerator:

\[
(2a + 3)(a - 1) = 2a^2 - 2a + 3a - 3 = 2a^2 + a - 3
\]

Thus, the right side becomes:

\[
\frac{2a^2 + a - 3 + 10}{(a - 1)(a + 1)} = \frac{2a^2 + a - 3 + 10}{(a - 1)(a + 1)} = \frac{2a^2 + a + 7}{(a - 1)(a + 1)}
\]

### Step 3: Set the numerators equal
Now that both sides have the same denominator, we can set the numerators equal:

\[
2a^2 - a - 3 = 2a^2 + a + 7
\]

### Step 4: Simplify the equation
Subtract \( 2a^2 \) from both sides:

\[
-a - 3 = a + 7
\]

Move all terms involving \( a \) to one side and constant terms to the other:

\[
-a - a = 7 + 3
\]

\[
-2a = 10
\]

Solve for \( a \):

\[
a = \frac{10}{-2} = -5
\]

### Solution:
The solution to the equation is \( a = -5 \).

Do you want more details or have any questions?

Here are five related questions:
1. How do you solve rational equations in general?
2. What are common factoring techniques for polynomials?
3. How do you handle equations with a difference of squares?
4. What other types of equations can be simplified with a common denominator?
5. How does multiplying both sides by a common denominator affect an equation?

**Tip:** When solving rational equations, always check for any restrictions in the domain (such as values that would make the denominator zero).

This page provides a step-by-step solution for solving the rational equation 2a - 3 over a - 1 equals 2a + 3 over a + 1 plus 10 over a² - 1. Learn how to factor difference of squares and find a common denominator to solve rational equations. Suitable for Grades 10-12 students.

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