Final answer:
To factor 3x³-4x²-28x-16 completely knowing (x+2) is a factor, long division must be used to divide the polynomial by x+2 to find the quotient, and then factor the resulting quadratic expression to get the fully factored form.
Step-by-step explanation:
To factor the polynomial 3x³-4x²-28x-16 completely given that (x+2) is a factor, we can use long division or synthetic division. Since we are instructed to use long division, we will do so. We begin by setting up the division, writing 3x³-4x²-28x-16 as the dividend and x+2 as the divisor.
During the division process, we find that 3x³ divided by x gives us 3x², which when multiplied by the divisor x+2 yields 3x³+6x². Subtracting this from the first two terms of the dividend, we get -10x²-28x. Continuing in this fashion, we determine the quotient to be 3x²-10x-8, and the remainder is zero since (x+2) is a factor.
Next, we factor the quadratic 3x²-10x-8 to obtain two binomials, which would provide the fully factored form of the original polynomial. If we find it difficult to factor the quadratic directly, we may use the quadratic formula or other factoring techniques.