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.