Final answer:
All provided options A, B, and C show correctly factored forms of quadratic expressions. We determined this by checking if the binomial factors multiply to give the original quadratic equation of form y^2 + by + c. Option D is not applicable since each of the other options can be factored successfully.
Step-by-step explanation:
To factor the quadratic expression completely, we need to find two numbers that multiply to give the constant term (the product of the coefficients of the quadratic equation) and also add up to give the coefficient of the middle term. The options provided all represent factored forms of quadratic expressions, so let's evaluate each option to see if it is correctly factored:
- A) (y + 1)(y + 7): Multiplying these factors gives y^2 + 8y + 7. To check if this is factored correctly, we look for two numbers that multiply to 7 and add up to 8, which are 1 and 7. So, option A is correctly factored.
- B) (y + 7)(y - 1): Multiplying gives y^2 + 6y - 7. For this expression to be correctly factored, we need two numbers that multiply to -7 and add up to 6, which are 7 and -1. So, option B is also correctly factored.
- C) (y + 2)(y + 5): Multiplying gives y^2 + 7y + 10. Here, we need two numbers that multiply to 10 and add up to 7, which are 2 and 5. Hence, option C is correctly factored.
- D) Cannot be factored: Since options A, B, and C are all correctly factored, option D does not apply here as all given expressions can be factored.
In conclusion, the original quadratic expression can indeed be factored, and options A, B, and C all represent correct factorizations of quadratic expressions.