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The problem appears to be simplifying this fraction:

$\frac{b^2 - 4b - 21}{b^2 - 15b + 50} - \frac{b + 3}{b - 10}$

Step 1: Factor the expressions

1. For $b^2 - 4b - 21$: Factor the quadratic expression: $b^2 - 4b - 21 = (b - 7)(b + 3)$

2. For $b^2 - 15b + 50$: Factor this quadratic as well: $b^2 - 15b + 50 = (b - 10)(b - 5)$

Thus, the first fraction becomes: $\frac{(b - 7)(b + 3)}{(b - 10)(b - 5)}$

Step 2: Simplify the second fraction

The second fraction is already simplified: $\frac{b + 3}{b - 10}$

Step 3: Combine the fractions

Both fractions have a common denominator $b - 10$, so rewrite the expression as: $\frac{(b - 7)(b + 3)}{(b - 10)(b - 5)} - \frac{b + 3}{b - 10}$

Multiply the numerator and denominator of the second fraction by $b - 5$ to get a common denominator: $\frac{b + 3}{b - 10} = \frac{(b + 3)(b - 5)}{(b - 10)(b - 5)}$

Now the expression becomes: $\frac{(b - 7)(b + 3)}{(b - 10)(b - 5)} - \frac{(b + 3)(b - 5)}{(b - 10)(b - 5)}$

Step 4: Combine the numerators

Since the denominators are the same, combine the numerators: $\frac{(b - 7)(b + 3) - (b + 3)(b - 5)}{(b - 10)(b - 5)}$

Factor out $b + 3$ from the numerator: $\frac{(b + 3)\left[(b - 7) - (b - 5)\right]}{(b - 10)(b - 5)}$

Simplify the term in brackets: $(b - 7) - (b - 5) = b - 7 - b + 5 = -2$

So the numerator becomes: $(b + 3)(-2)$

Thus, the fraction is now: $\frac{-2(b + 3)}{(b - 10)(b - 5)}$

Final Answer:

$\frac{-2(b + 3)}{(b - 10)(b - 5)}$

Let me know if you need further clarification or more details! Here are some related questions:

1. How do you factorize quadratic expressions like $b^2 - 4b - 21$?
2. How can you simplify fractions with polynomials in the numerator and denominator?
3. What is the importance of a common denominator when subtracting fractions?
4. How do you know when to factor out terms from a numerator?
5. Can this simplified fraction be reduced further under specific conditions?

Tip: Always check for common factors across numerators and denominators before finalizing your answer.