To multiply the two fractions
\[ \frac{8s^2 - 57s + 7}{8s^2 + 63s - 8} \times \frac{7s^2 + 55s - 8}{56s^2 - 15s + 1}, \]
we can first focus on factoring each polynomial.
-
Factoring the numerator \( 8s^2 - 57s + 7 \): \[ 8s^2 - 57s + 7 = (8s + 1)(s - 7) \] (This can be verified via expansion)
-
Factoring the denominator \( 8s^2 + 63s - 8 \): \[ 8s^2 + 63s - 8 = (8s - 1)(s + 8) \] (This can also be verified via expansion)
-
Factoring the numerator \( 7s^2 + 55s - 8 \): \[ 7s^2 + 55s - 8 = (7s - 1)(s + 8) \] (Verification via expansion is recommended)
-
Factoring the denominator \( 56s^2 - 15s + 1 \): \[ 56s^2 - 15s + 1 = (7s - 1)(8s - 1) \] (Again, verification via expansion is recommended)
Now substituting all of these factors into the product gives:
\[ \frac{(8s + 1)(s - 7)}{(8s - 1)(s + 8)} \times \frac{(7s - 1)(s + 8)}{(7s - 1)(8s - 1)} \]
Next, we cancel the common factors:
- We see \( (7s - 1) \) cancels from the numerator and denominator.
- \( (s + 8) \) cancels from the numerator and denominator.
- \( (8s - 1) \) cancels from the numerator and denominator.
After cancellation, we are left with:
\[ \frac{(8s + 1)(s - 7)}{1}. \]
This simplifies to:
\[ (8s + 1)(s - 7) \]
Expanding this product gives:
\[ 8s^2 - 56s + s - 7 = 8s^2 - 55s - 7. \]
Thus, the answer is:
\[ 8s^2 - 55s - 7. \]
The simplified form is \(\boxed{8s^2 - 55s - 7}\).