Final answer:
To check if x+2 is a factor of P(x)=2x³+5x²+5x+6, we plug x=-2 into P(x). Since P(-2)=0, x+2 is a factor, and through division, we find that P(x) can be written as (x+2)(2x²-x+3).
Step-by-step explanation:
To determine if x+2 is a factor of the polynomial P(x)=2x³+5x²+5x+6, we can use the Factor Theorem, which tells us that if x=c is a root of the polynomial P(x), then x-c is a factor of P(x). Plugging in x=-2 into P(x), we get:
P(-2) = 2(-2)³ + 5(-2)² + 5(-2) + 6 = -16 + 20 - 10 + 6 = 0
Since P(-2) = 0, x+2 is indeed a factor of P(x). To write P(x) as a product of two factors, we can perform polynomial division or use a method like synthetic division to divide P(x) by x+2, obtaining the other factor.
If you perform the division, the result will be P(x) = (x+2)(2x²-x+3).