How do I divide (2 + 3 - 3x² + 2x - 4) + (x - 3)?

To divide the expression (2 + 3 - 3x² + 2x - 4) by (x - 3), you'll go through the following steps:

1. Set up the division: Write the dividend (2 + 3 - 3x² + 2x - 4) and the divisor (x - 3) in the long division format. However, the provided expression seems to combine terms from the dividend and divider improperly. The operation between the two polynomials should be specified. Assuming the "+" between them is an indication to add them before dividing, let’s first combine like terms: (2 + 3 - 3x² + 2x - 4) + (x - 3) simplifies to: -3x² + 3x + 1 [Because 2 + 3 - 4 = 1 and 2x + x = 3x]

2. Arrange in descending power of x: Write the terms of both the dividend and the divisor in decreasing powers of x even if some terms are missing.

For the dividend, -3x² + 3x + 1.

For the divisor, x - 3.

3. Divide the leading terms: Take the leading term of the dividend (-3x²) and divide it by the leading term of the divisor (x), which gives -3x. Write this above the division line.

4. Multiply the divisor by this term: Multiply (x - 3) by -3x, which gives -3x² + 9x. Write this below the -3x² + 3x + 1.

5. Subtract: Subtract (-3x² + 9x) from -3x² + 3x + 1.

(-3x² + 3x + 1) -(-3x² + 9x)

The -3x² terms cancel out, and you're left with -6x + 1.

6. Bring down the next term if applicable, and repeat: In this case, there are no more terms to bring down. You would now repeat the division process with -6x + 1 as your new dividend, but since the degree of the new dividend (the power of x) is less than the degree of the divisor, this term is actually the remainder of the division.

7. Write out the final answer in the form of quotient plus remainder/divisor: The quotient is -3x, and the remainder is -6x + 1. Your final answer can be expressed as:

-3x + (-6x + 1)/(x - 3).

There is no further simplification of this expression since the remainder is already in its simplest form.