Let's start by solving the given equation:

\[
\frac{3m}{m - 5} - \frac{2}{m + j} = \frac{2m^2 + 8m}{m^2 - 4m - 5}
\]

### Step 1: Simplify the right-hand side

We can factor the quadratic in the denominator of the right-hand side:

\[
m^2 - 4m - 5 = (m - 5)(m + 1)
\]

So the equation becomes:

\[
\frac{3m}{m - 5} - \frac{2}{m + j} = \frac{2m^2 + 8m}{(m - 5)(m + 1)}
\]

### Step 2: Simplify the numerator on the right-hand side

Factor the numerator \( 2m^2 + 8m \):

\[
2m^2 + 8m = 2m(m + 4)
\]

Thus, the equation becomes:

\[
\frac{3m}{m - 5} - \frac{2}{m + j} = \frac{2m(m + 4)}{(m - 5)(m + 1)}
\]

### Step 3: Find a common denominator on the left-hand side

The first term on the left has the denominator \( m - 5 \) and the second term has \( m + j \). We can combine the left-hand side by finding a common denominator, but it's challenging without knowing the value of \( j \).

At this point, we would need more information or assumptions regarding the value of \( j \) to proceed with solving for \( m \).

Do you want to continue under the assumption that \( j \) is a constant? Or perhaps make any other clarification?

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### Related questions:
1. How do you handle equations with unknown parameters like \( j \)?
2. How would the solution change if \( j = 1 \)?
3. What are some factoring techniques for complex polynomials?
4. How can you check the domain of rational expressions with unknown variables?
5. What happens when \( m = 5 \) or \( m = -1 \) in this equation?

**Tip:** Always look for factorization opportunities to simplify complex algebraic expressions.

Explore how to solve a complex rational equation involving factoring quadratic expressions. This detailed step-by-step guide covers algebraic techniques such as factoring, simplifying rational expressions, and finding common denominators. Suitable for students in Grades 10-12.

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