We need to simplify the given algebraic expression:

$\frac{m-3}{m-2} + \frac{7-4m}{2m^2-9m+10} - \frac{m+1}{2m-5}$

Step 1: Factor the denominator in the second fraction

The denominator of the second fraction is:

$2m^2 - 9m + 10$

We factor this quadratic expression:

$2m^2 - 9m + 10 = (2m - 5)(m - 2)$

So, the expression becomes:

$\frac{m-3}{m-2} + \frac{7-4m}{(2m-5)(m-2)} - \frac{m+1}{2m-5}$

Step 2: Rewrite $7 - 4m$

Rewriting $7 - 4m$ to match factored terms:

$7 - 4m = - (4m - 7)$

Thus:

$\frac{-(4m-7)}{(2m-5)(m-2)}$

Step 3: Find the Common Denominator

The least common denominator (LCD) is $(2m-5)(m-2)$, so we rewrite each fraction:

1. $\frac{m-3}{m-2} = \frac{(m-3)(2m-5)}{(m-2)(2m-5)}$
2. $\frac{-(4m-7)}{(2m-5)(m-2)}$ stays the same.
3. $\frac{m+1}{2m-5} = \frac{(m+1)(m-2)}{(2m-5)(m-2)}$

Step 4: Combine the Numerators

$\frac{(m-3)(2m-5) - (4m-7) - (m+1)(m-2)}{(m-2)(2m-5)}$

Expanding each term:

1. $(m-3)(2m-5) = 2m^2 - 5m - 6m + 15 = 2m^2 - 11m + 15$
2. $-(4m - 7) = -4m + 7$
3. $(m+1)(m-2) = m^2 - 2m + m - 2 = m^2 - m - 2$

Now, combine:

$2m^2 - 11m + 15 - 4m + 7 - (m^2 - m - 2)$

$2m^2 - 11m + 15 - 4m + 7 - m^2 + m + 2$

$(2m^2 - m^2) + (-11m - 4m + m) + (15 + 7 + 2)$

$m^2 - 14m + 24$

Step 5: Factor $m^2 - 14m + 24$

$m^2 - 14m + 24 = (m-12)(m-2)$

Step 6: Simplify

$\frac{(m-12)(m-2)}{(m-2)(2m-5)}$

Cancel out $(m-2)$:

$\frac{m-12}{2m-5}$

Final Answer:

$\frac{m-12}{2m-5}$

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Would you like a step-by-step breakdown of any part?

Here are 5 related questions to explore:

1. How do you find the least common denominator (LCD) of algebraic fractions?
2. How do you factor quadratic expressions like $2m^2 - 9m + 10$?
3. What happens when common terms cancel in a fraction?
4. How do you simplify algebraic fractions with multiple terms?
5. What is the importance of factoring in algebraic simplifications?

Tip: Always check for common factors before attempting to simplify fractions—it makes the process much easier!