Final answer:
The quadratic expression 3a^(2)+18a+15 is factored by first finding two numbers that multiply to 45 (the product of 3 and 15) and add to 18. The expression is then rewritten as 3a^2 + 3a + 15a + 15, factored by grouping to 3(a + 5)(a + 1), resulting in the fully factored form: 3(a + 5)(a + 1).
Step-by-step explanation:
To factor the expression 3a^(2)+18a+15, we need to find two numbers that multiply to give the product of the coefficient of a^2 (which is 3) and the constant term (which is 15), and at the same time, sum up to give the coefficient of the linear term a (which is 18). The two numbers that meet these conditions are 3 and 15 themselves.
Thus, we can write:
3a^2 + 18a + 15
= 3a^2 + 3a + 15a + 15
= 3a(a + 1) + 15(a + 1)
= (3a + 15)(a + 1)
Finally, we can factor out a 3 from (3a + 15):
= 3(a + 5)(a + 1)
And we end up with the completely factored expression: 3(a + 5)(a + 1).