Final answer:
To determine the common denominator, factor the denominators and find the LCM.
Step-by-step explanation:
To determine the common denominator needed to simplify the expression, we need to find the least common multiple (LCM) of the denominators. The denominators are (4x²-3x-1) and (4x²+7x-2). We can factor both denominators to simplify the process.
Factoring the first denominator, we get: (4x²-3x-1) = (2x+1)(2x-1)
Factoring the second denominator, we get: (4x²+7x-2) = (4x-1)(x+2)
The LCM is the product of the highest powers of all the prime factors involved. In this case, the LCM is (2x+1)(2x-1)(4x-1)(x+2). Therefore, the common denominator needed to simplify the expression is (2x+1)(2x-1)(4x-1)(x+2).