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Let's break down and simplify the given expression step by step.

### The expression to simplify is:
\[
\frac{2x^2 - 9x + 10}{(x - 2)^3 + (x - 3)^3} \div \frac{1}{x^2 - 5x + 7}
\]

### Step 1: Simplify the denominator on the left-hand side:
We have:
\[
(x - 2)^3 + (x - 3)^3
\]

Use the identity for the sum of cubes:
\[
a^3 + b^3 = (a + b)(a^2 - ab + b^2)
\]

Here, \(a = (x - 2)\) and \(b = (x - 3)\). So, applying the formula:

1. \(a + b = (x - 2) + (x - 3) = 2x - 5\)
2. \(a^2 - ab + b^2 = (x - 2)^2 - (x - 2)(x - 3) + (x - 3)^2\)

Now let's compute each part:

- \((x - 2)^2 = x^2 - 4x + 4\)
- \((x - 3)^2 = x^2 - 6x + 9\)
- \((x - 2)(x - 3) = x^2 - 5x + 6\)

Now combine these:

\[
(x - 2)^2 - (x - 2)(x - 3) + (x - 3)^2 = (x^2 - 4x + 4) - (x^2 - 5x + 6) + (x^2 - 6x + 9)
\]
\[
= x^2 - 4x + 4 - x^2 + 5x - 6 + x^2 - 6x + 9 = x^2 - 5x + 7
\]

Thus, the denominator becomes:
\[
(2x - 5)(x^2 - 5x + 7)
\]

### Step 2: Simplify the entire expression
Now, rewrite the original expression with the simplified denominator:

\[
\frac{2x^2 - 9x + 10}{(2x - 5)(x^2 - 5x + 7)} \div \frac{1}{x^2 - 5x + 7}
\]

Dividing by a fraction is the same as multiplying by its reciprocal:

\[
= \frac{2x^2 - 9x + 10}{(2x - 5)(x^2 - 5x + 7)} \times (x^2 - 5x + 7)
\]

The \(x^2 - 5x + 7\) terms cancel out, leaving:

\[
\frac{2x^2 - 9x + 10}{2x - 5}
\]

### Step 3: Factor the numerator
Now factor \(2x^2 - 9x + 10\). We need to find factors of 20 that add up to -9. These are -5 and -4:

\[
2x^2 - 9x + 10 = (2x - 5)(x - 2)
\]

So, the expression becomes:
\[
\frac{(2x - 5)(x - 2)}{2x - 5}
\]

Cancel the \(2x - 5\) terms:
\[
x - 2
\]

Thus, the simplified expression is **\(x - 2\)**.

### Final Answer:
\[
\boxed{x - 2}
\]

Let me know if you need any further details or have questions!

Here are 5 related questions you might find interesting:
1. What is the sum of cubes identity used for more complex expressions?
2. How do you factor trinomials like \(2x^2 - 9x + 10\) efficiently?
3. What are the restrictions of this expression, and why do we cancel terms carefully?
4. Can the expression be simplified further in specific cases if there were different terms in the denominator?
5. How would you handle this expression if it were a more complex rational equation?

**Tip**: Always check for common factors between numerators and denominators before multiplying or dividing rational expressions—it often simplifies the process!

Considerando las restricciones del caso, al simplificar la expresión algebraica: (2x^2 - 9x + 10) / ((x - 2)^3 + (x - 3)^3) ÷ (1 / (x^2 - 5x + 7)), se obtiene:

Simplify the algebraic expression (2x^2 - 9x + 10) / ((x - 2)^3 + (x - 3)^3) ÷ (1 / (x^2 - 5x + 7)). This solution involves the sum of cubes identity and factoring quadratic trinomials. Suitable for high school students in Grades 10-12.

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