Let's break down and solve each part of the question step by step.

(a) Finding the values of $m$ and $n$

Given that the GCD of $f(x)$ and $g(x)$ is $x^2 + x - 2$, we need to find the values of $m$ and $n$ for the polynomials:

$f(x) = x^3 + 5x^2 - 2mx - 4n$ $g(x) = nx^3 - mx^2 - 5x + 2$

Step 1: Factorize the GCD
The given GCD is $x^2 + x - 2$. We factor it:

$x^2 + x - 2 = (x - 1)(x + 2)$

Thus, the GCD of $f(x)$ and $g(x)$ is $(x - 1)(x + 2)$.

Step 2: Polynomial Division
To solve for $m$ and $n$, we would divide both $f(x)$ and $g(x)$ by $x^2 + x - 2$. Let's perform polynomial division for both $f(x)$ and $g(x)$.

We will divide $f(x) = x^3 + 5x^2 - 2mx - 4n$ by $x^2 + x - 2$ and $g(x) = nx^3 - mx^2 - 5x + 2$ by $x^2 + x - 2$, which will allow us to match coefficients with the GCD structure.

Division of $f(x)$:

1. Divide the highest degree term of $f(x)$, which is $x^3$, by the highest degree term of the divisor $x^2$, resulting in $x$.
2. Multiply $x$ by $x^2 + x - 2$ and subtract the result from $f(x)$.
3. Repeat the process with the remainder.

Division of $g(x)$:

1. Divide the highest degree term of $g(x)$, which is $nx^3$, by the highest degree term of the divisor $x^2$, resulting in $nx$.
2. Multiply $nx$ by $x^2 + x - 2$ and subtract the result from $g(x)$.
3. Repeat the process with the remainder.

This method will yield the values of $m$ and $n$ by matching the coefficients. After performing this, we'll compare the remainders and the structure of the divisions.

(b) Finding the Least Common Multiple (LCM)

The relationship between the GCD and LCM of two polynomials is given by:

$\text{LCM}(f(x), g(x)) = \frac{f(x)g(x)}{\text{GCD}(f(x), g(x))}$

From part (a), we know that the GCD is $x^2 + x - 2$, so:

$\text{LCM}(f(x), g(x)) = \frac{f(x)g(x)}{x^2 + x - 2}$

You can substitute the expressions for $f(x)$ and $g(x)$ and perform polynomial multiplication of $f(x)$ and $g(x)$, then divide by $x^2 + x - 2$ to find the LCM.

(c) Simplifying the Given Expression

We are given the expression:

$\frac{3x + 10}{x^2 + 3x - 4}$

We want to simplify this expression. To do so, we factor the denominator.

1. Factor the quadratic expression $x^2 + 3x - 4$: $x^2 + 3x - 4 = (x - 1)(x + 4)$

Now the expression becomes:

$\frac{3x + 10}{(x - 1)(x + 4)}$

At this stage, the expression is already simplified, as there are no common factors between the numerator and the denominator.

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Summary of Steps for Each Part:

1. (a) Find the values of $m$ and $n$ by performing polynomial division and matching coefficients with the GCD.
2. (b) Calculate the LCM using the formula $\text{LCM}(f(x), g(x)) = \frac{f(x)g(x)}{\text{GCD}(f(x), g(x))}$.
3. (c) Simplify the given rational expression by factoring the denominator and checking for common factors.

Would you like to see the detailed calculations for the division and factorization process, or do you have any questions about the steps?

Here are some related questions:

1. How do you perform polynomial long division?
2. How do you compute the GCD of two polynomials?
3. What is the relationship between the roots of a polynomial and its factors?
4. How do you find the degree of a polynomial product?
5. What is the significance of the LCM and GCD in algebraic expressions?

Tip: When performing polynomial division, always match the highest degree term of the dividend and divisor first.