Explanation:
If (x + 2) is a factor of x³ + 2x² + 3x + 6, then we know that dividing the polynomial by (x + 2) will result in a quotient polynomial of degree 2 and a remainder of 0.
We can use long division or synthetic division to find the quotient polynomial, but here we will use long division:
x² + 1x + 3
---------------------
x + 2 | x³ + 2x² + 3x + 6
x³ + 2x²
---------
1x² + 3x
1x² + 2x
--------
1x + 6
1x + 2
-----
4
Therefore,
x³ + 2x² + 3x + 6 = (x + 2)(x² + x + 3) + 4
So the factored form of the polynomial is:
x³ + 2x² + 3x + 6 = (x + 2)(x² + x + 3) + 4
where (x + 2) is a factor and the other factor is x² + x + 3.