nothetic division to divide. \( 2 x^{2}-x \) by \( (x-2) \) the distance between the given points.

Divide the polynomials by following steps: - step0: Using Long Division to Divide Polynomials: \(\left(2x^{2}-x\right)\div \left(x-2\right)\) - step1: Calculate: \(\left(2x+3\right)\text{ }\textrm{remainder}\text{ }6\) Let's solve the two parts of your request step by step. ### Part 1: Polynomial Long Division We need to divide \( 2x^2 - x \) by \( x - 2 \). 1. **Set up the division**: - Dividend: \( 2x^2 - x \) - Divisor: \( x - 2 \) 2. **Perform the division**: - Divide the leading term of the dividend \( 2x^2 \) by the leading term of the divisor \( x \) to get \( 2x \). - Multiply \( 2x \) by the entire divisor \( (x - 2) \) to get \( 2x^2 - 4x \). - Subtract this from the original dividend: \[ (2x^2 - x) - (2x^2 - 4x) = 3x \] - Now, bring down the next term (if any). In this case, there are no more terms to bring down. 3. **Repeat the process**: - Divide \( 3x \) by \( x \) to get \( 3 \). - Multiply \( 3 \) by \( (x - 2) \) to get \( 3x - 6 \). - Subtract: \[ 3x - (3x - 6) = 6 \] 4. **Final result**: - The quotient is \( 2x + 3 \) and the remainder is \( 6 \). - Therefore, the result of the division is: \[ 2x + 3 \text{ remainder } 6 \] ### Part 2: Distance Between Given Points It seems there was a misunderstanding regarding the distance between points, as no specific points were provided. If you have two points, say \( (x_1, y_1) \) and \( (x_2, y_2) \), the distance \( d \) between them can be calculated using the formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Please provide the coordinates of the points you want to calculate the distance between, and I can help you with that!