Final answer:
To find the quotient of the given rational expressions, one must multiply the first fraction by the reciprocal of the second, factor the polynomials if possible, and then simplify by canceling out any common terms.
Step-by-step explanation:
The student has posed a question concerning dividing one rational expression by another. The expressions involve quadratic polynomials. The division of two fractions is equivalent to multiplying the first fraction by the reciprocal of the second. Thus, the problem 4x²−9 / (6x²+13x+6) ÷ (4x²−1) / (6x²+x−2) simplifies to (4x²−9) / (6x²+13x+6) ⋅ (6x²+x−2) / (4x²−1). To proceed, we should factor the polynomials where possible and then multiply the numerators and denominators accordingly. If the polynomials factor to reveal any common terms, those may cancel out, simplifying the expression into its simplest form.
To solve this, we need to use algebraic techniques such as factoring and applying the properties of exponents. The mention of negative exponents indicates that they should be moved to the denominator as part of the algebraic manipulation process. Upon completing these steps and simplifying fully, you'll obtTo find the quotient and write the answer in simplest form, we need to simplify and divide the given expressions.First, simplify the expression in the numerator:4x²−9 / 6x²+13x+Next, simplify the expression in the denominator:4x²−1 / 6x²+x−2After simplifying both expressions, divide the numerator by the denominator to find the quotient.ain the final quotient in its simplest form.