Final answer:
The quadratic expression 5k^2 + 30k + 45 can be factored by first taking out the common factor of 5 and then factoring the resulting trinomial, which simplifies to 5(k + 3)^2, corresponding to Option B.
Step-by-step explanation:
The task is to factor the quadratic expression 5k^2 + 30k + 45. We first notice that there is a common factor of 5 in all terms. Factoring out the 5, we get:
5k^2 + 30k + 45 = 5(k^2 + 6k + 9)
Next, we look for two numbers that multiply to 9 (the constant term) and add up to 6 (the coefficient of the middle term). In this case, both numbers are 3, so the quadratic expression k^2 + 6k + 9 can be factored into:
k^2 + 6k + 9 = (k + 3)(k + 3) or (k + 3)^2
Therefore, the factored form of 5k^2 + 30k + 45 is:
5(k + 3)^2
So, the correct answer to the factorization problem is Option B: 5(k+3)^2.