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The given expression is:

\[
\frac{-5}{x^2 - 9x + 8} + \frac{6}{x^2 - 10x + 16}
\]

Let's simplify it step by step.

### Step 1: Factor the Denominators

#### First denominator: \( x^2 - 9x + 8 \)
We look for two numbers that multiply to \(8\) and add to \(-9\). These numbers are \(-1\) and \(-8\). So, the factorization is:

\[
x^2 - 9x + 8 = (x - 1)(x - 8)
\]

#### Second denominator: \( x^2 - 10x + 16 \)
We need two numbers that multiply to \(16\) and add to \(-10\). These numbers are \(-4\) and \(-4\), so the factorization is:

\[
x^2 - 10x + 16 = (x - 4)^2
\]

### Step 2: Rewrite the expression with factored denominators
Now, the expression becomes:

\[
\frac{-5}{(x - 1)(x - 8)} + \frac{6}{(x - 4)^2}
\]

### Step 3: Find the Least Common Denominator (LCD)
The LCD is the product of all distinct factors from the denominators:

\[
\text{LCD} = (x - 1)(x - 8)(x - 4)^2
\]

### Step 4: Rewrite each fraction with the LCD as the denominator
1. Multiply the first fraction by \(\frac{(x - 4)^2}{(x - 4)^2}\):

\[
\frac{-5}{(x - 1)(x - 8)} = \frac{-5(x - 4)^2}{(x - 1)(x - 8)(x - 4)^2}
\]

2. The second fraction already has \((x - 4)^2\) in its denominator, so multiply by \(\frac{(x - 1)(x - 8)}{(x - 1)(x - 8)}\):

\[
\frac{6}{(x - 4)^2} = \frac{6(x - 1)(x - 8)}{(x - 1)(x - 8)(x - 4)^2}
\]

### Step 5: Combine the fractions
Now that both fractions have the same denominator, we can combine them:

\[
\frac{-5(x - 4)^2 + 6(x - 1)(x - 8)}{(x - 1)(x - 8)(x - 4)^2}
\]

### Step 6: Expand and simplify the numerator
1. Expand \(-5(x - 4)^2\):

\[
-5(x - 4)^2 = -5(x^2 - 8x + 16) = -5x^2 + 40x - 80
\]

2. Expand \(6(x - 1)(x - 8)\):

\[
(x - 1)(x - 8) = x^2 - 9x + 8
\]

\[
6(x^2 - 9x + 8) = 6x^2 - 54x + 48
\]

3. Combine both expanded expressions:

\[
-5x^2 + 40x - 80 + 6x^2 - 54x + 48 = (6x^2 - 5x^2) + (40x - 54x) + (-80 + 48)
\]

\[
= x^2 - 14x - 32
\]

### Final Result
The simplified form of the expression is:

\[
\frac{x^2 - 14x - 32}{(x - 1)(x - 8)(x - 4)^2}
\]

Let me know if you'd like further details!

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Here are some follow-up questions to explore the problem further:

1. How would you factor the numerator \(x^2 - 14x - 32\) further, if possible?
2. What happens to the expression at \(x = 1\), \(x = 4\), or \(x = 8\)? Are there any restrictions?
3. How does simplifying the numerator change the nature of the function (zeros, asymptotes, etc.)?
4. Could you expand this to consider limits as \(x\) approaches different critical points?
5. How would the graph of this rational function look like?

**Tip:** Always check for possible restrictions by considering the values of \(x\) that make the denominator zero, as these will represent vertical asymptotes or undefined points.

This problem explains how to simplify the rational expression involving -5/(x^2 - 9x + 8) + 6/(x^2 - 10x + 16) by factoring the quadratic denominators and finding the least common denominator. Suitable for high school students studying algebra.

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