To multiply the two fractions, we'll first express it as:
\[ \frac{8s^2 - 57s + 7}{8s^2 + 63s - 8} \times \frac{7s^2 + 55s - 8}{56s^2 - 15s + 1} \]
Next, we'll multiply the numerators together and the denominators together:
\[ \frac{(8s^2 - 57s + 7)(7s^2 + 55s - 8)}{(8s^2 + 63s - 8)(56s^2 - 15s + 1)} \]
Now, we need to simplify both the numerator and the denominator by either factoring them or directly performing the multiplication.
Step 1: Multiply the Numerator
Numerator:
\[ (8s^2 - 57s + 7)(7s^2 + 55s - 8) \]
Using the distributive property (FOIL method):
- \( 8s^2 \cdot 7s^2 = 56s^4 \)
- \( 8s^2 \cdot 55s = 440s^3 \)
- \( 8s^2 \cdot (-8) = -64s^2 \)
- \( -57s \cdot 7s^2 = -399s^3 \)
- \( -57s \cdot 55s = -3135s^2 \)
- \( -57s \cdot (-8) = 456s \)
- \( 7 \cdot 7s^2 = 49s^2 \)
- \( 7 \cdot 55s = 385s \)
- \( 7 \cdot (-8) = -56 \)
Combining all terms:
\[ 56s^4 + (440s^3 - 399s^3) + (-64s^2 - 3135s^2 + 49s^2) + (456s + 385s) - 56 \]
This simplifies to:
\[ 56s^4 + 41s^3 - 3150s^2 + 841s - 56 \]
Step 2: Multiply the Denominator
Denominator:
\[ (8s^2 + 63s - 8)(56s^2 - 15s + 1) \]
Using the distributive property:
- \( 8s^2 \cdot 56s^2 = 448s^4 \)
- \( 8s^2 \cdot (-15s) = -120s^3 \)
- \( 8s^2 \cdot 1 = 8s^2 \)
- \( 63s \cdot 56s^2 = 3528s^3 \)
- \( 63s \cdot (-15s) = -945s^2 \)
- \( 63s \cdot 1 = 63s \)
- \( -8 \cdot 56s^2 = -448s^2 \)
- \( -8 \cdot (-15s) = 120s \)
- \( -8 \cdot 1 = -8 \)
Combining all terms:
\[ 448s^4 + (-120s^3 + 3528s^3) + (8s^2 - 945s^2 - 448s^2) + (63s + 120s) - 8 \]
This simplifies to:
\[ 448s^4 + 3408s^3 - 1385s^2 + 183s - 8 \]
Final Result
Combining the results into a single fraction gives us:
\[ \frac{56s^4 + 41s^3 - 3150s^2 + 841s - 56}{448s^4 + 3408s^3 - 1385s^2 + 183s - 8} \]
This is the product of the two original fractions in its simplified form. Further factoring may be done if possible, but this is the multiplication result.