simplify (b ^ 2 - 9b + 14)/(b ^ 2 - 4b - 21) / ((b ^ 2 + 2b - 8)/(9 + 3b))
To simplify the expression
```
(b^2 - 9b + 14)/(b^2 - 4b - 21) / ((b^2 + 2b - 8)/(9 + 3b))
```
we can start by factoring the numerators and the denominators.
The numerator (b^2 - 9b + 14) can be factored as:
(b^2 - 9b + 14) = (b - 7)(b - 2)
The denominator (b^2 - 4b - 21) can be factored as:
(b^2 - 4b - 21) = (b - 7)(b + 3)
The second numerator (b^2 + 2b - 8) can be factored as:
(b^2 + 2b - 8) = (b + 4)(b - 2)
Now we substitute the factored values back into the expression:
[(b - 7)(b - 2)]/[(b - 7)(b + 3)] / [(b + 4)(b - 2)]/(9 + 3b)
Next, we can cancel out common factors in the numerator and the denominator.
The (b - 2) term cancels out in the numerator and the denominator, and the (b - 7) term cancels out as well:
[(b - 7)(b - 2)]/[(b - 7)(b + 3)] / [(b + 4)]/(9 + 3b)
Now, we can simplify further by dividing the fractions. Dividing by a fraction is the same as multiplying by its reciprocal:
[(b - 7)(b - 2)]/[(b - 7)(b + 3)] * [(9 + 3b)/(b + 4)]
We can cancel out the (b - 7) term from the numerator and the denominator:
[(b - 2)]/(b + 3) * [(9 + 3b)/(b + 4)]
Finally, we can simplify the expression by multiplying the remaining terms:
[(b - 2)(9 + 3b)]/(b + 3)(b + 4)
So, the simplified expression is:
```
(b - 2)(9 + 3b) / (b + 3)(b + 4)
```