What is the remainder when 3x³ + 2x² + 4x + 3 is divided by x² + 3x + 3?

To find the remainder when dividing the polynomial 3x³ + 2x² + 4x + 3 by x² + 3x + 3, we can use polynomial long division or synthetic division.

First, we divide 3x³ by x², which gives 3x. We then multiply 3x by the entire divisor x² + 3x + 3:

3x * (x² + 3x + 3) = 3x³ + 9x² + 9x

Now we subtract this from our original polynomial:

(3x³ + 2x² + 4x + 3) - (3x³ + 9x² + 9x) = -7x² - 5x + 3

Next, we divide -7x² by x², which gives us -7. We multiply -7 by the divisor:

-7 * (x² + 3x + 3) = -7x² - 21x - 21

We subtract this again:

(-7x² - 5x + 3) - (-7x² - 21x - 21) = 16x + 24

Since we have a polynomial of degree 1 (16x + 24) which is less than the degree of the divisor (degree 2), we cannot divide any further. Therefore, the remainder is:

16x + 24.

In conclusion, when 3x³ + 2x² + 4x + 3 is divided by x² + 3x + 3, the remainder is 16x + 24.