Final answer:
The polynomial 5x² + 5x + 60 can be factored by first taking out the common factor of 5, but cannot be factored further into two binomials since there are no two integers that multiply to 12 and add up to 1. None of the provided answer choices are correct.
Step-by-step explanation:
To factor the polynomial completely and factor out the greatest common factor, we need to first identify any common factors in the coefficients of the terms. The polynomial given is 5x² + 5x + 60. We can see that each term has a factor of 5, so we begin by factoring out the 5:
5(x² + x + 12).
Now we need to factor the quadratic expression inside the parenthesis. However, there are no two numbers that multiply to 12 and add up to 1, which is what we would need to factor it further as a quadratic expression. Therefore, the expression does not factor further and the fully factored form of the polynomial is:
5(x² + x + 12)
None of the answer choices provided match this factored form, so the correct factorization is not listed among the options supplied by the student.