Final answer:
To divide two algebraic fractions, multiply the first by the reciprocal of the second and simplify. Factor the numerators, take the reciprocal of the second fraction, and cancel common factors for the simplest quotient.
Step-by-step explanation:
When you are asked to divide two algebraic fractions and express the quotient in simplest form, what you are doing is essentially multiplying the first fraction by the reciprocal of the second. In this case, the division problem is expressed as:
\[((x^2 - x - 12) / (2x)) \div ((x^2 + x - 6) / (6x^2))\]
To solve this, find the inverses of the fractions where applicable and then simplify the equations:
- Factor both the numerators (if possible).
- Multiply the first fraction by the reciprocal (inverse) of the second fraction.
- Simplify the resulting expression by cancelling out any common factors from the numerator and the denominator.
The simplified form of this division will be the quotient we're looking to obtain.