Use polynomial long division to divide

12x3 – 9x2 + 13x + 4
by
3x2 – 3x + 4

To divide the polynomial 12x^3 – 9x^2 + 13x + 4 by 3x^2 – 3x + 4 using polynomial long division, follow these steps:

Step 1: Arrange the polynomials in descending powers of x, with missing terms filled with zero coefficients:
12x^3 – 9x^2 + 13x + 4 ÷ 3x^2 – 3x + 4

Step 2: Divide the leading term of the dividend by the leading term of the divisor:
12x^3 ÷ 3x^2

The result is 4x.

Step 3: Multiply the entire divisor by the result obtained in Step 2:
4x * (3x^2 – 3x + 4)

The product is 12x^3 – 12x^2 + 16x.

Step 4: Subtract the product from the original dividend:
(12x^3 – 9x^2 + 13x + 4) - (12x^3 – 12x^2 + 16x)

Simplifying the above expression gives:
-9x^2 + 13x + 4 - (-12x^2 + 16x)

Rearranging terms, the expression becomes:
-9x^2 + 13x + 4 + 12x^2 - 16x

The simplified expression is:
3x^2 - 3x + 4.

Step 5: Repeat the process with the simplified expression obtained in Step 4.
(3x^2 - 3x + 4) ÷ (3x^2 – 3x + 4)

Step 6: Divide the leading term of the dividend by the leading term of the divisor:
3x^2 ÷ 3x^2

The result is 1.

Step 7: Multiply the entire divisor by the result obtained in Step 6:
1 * (3x^2 – 3x + 4)

The product is 3x^2 – 3x + 4.

Step 8: Subtract the product from the simplified expression obtained in Step 5:
(3x^2 - 3x + 4) - (3x^2 – 3x + 4)

Simplifying the above expression gives:
-3x + 4 - (-3x + 4)

Rearranging terms, the expression becomes:
-3x + 4 + 3x - 4

The simplified expression is:
0.

Step 9: We have a remainder of zero, indicating that the division is exact. Therefore, the final result is the quotient obtained in Step 2, which is 4x.

So, 12x^3 – 9x^2 + 13x + 4 divided by 3x^2 – 3x + 4 equals 4x.