Final answer:
To simplify (x²-2x-37)/(x²-3x-40), factor both the numerator and denominator and cancel the common factor. The equivalent expression is (x - 37)/(x - 8).
Step-by-step explanation:
The task is to find an equivalent expression for the fraction (x²-2x-37)/(x²-3x-40). To do this, we need to factor both the numerator and the denominator. Factoring a quadratic involves finding two numbers that multiply to give the product of the coefficient of x² and the constant term (last number), and add to give the coefficient of the x term.
For the numerator, x² - 2x - 37, we are looking for two numbers that multiply to -37 and add to -2. These numbers are -5 and 37. So we can factor the numerator to (x - 37)(x + 5).
For the denominator, x² - 3x - 40, we are looking for two numbers that multiply to -40 and add to -3. These numbers are -8 and 5. So we can factor the denominator to (x - 8)(x + 5).
Now, we notice that (x + 5) appears in both the numerator and denominator, so we can cancel out this common factor. The simplified expression is (x - 37)/(x - 8).