Divide. \[ \left(x^{2}+2\right) \div(x-1) \]

To divide \( \left(x^{2}+2\right) \) by \( (x-1) \), we can use polynomial long division. 1. **Set up the division**: - Divide the leading term of the dividend \( x^2 \) by the leading term of the divisor \( x \), which gives us \( x \). 2. **Multiply**: - Multiply \( x \) by the entire divisor \( (x-1) \): \[ x(x - 1) = x^2 - x. \] 3. **Subtract**: - Now, subtract that result from the original polynomial: \[ (x^2 + 2) - (x^2 - x) = 0 + x + 2 = x + 2. \] 4. **Repeat the process**: - Now, we need to divide \( (x + 2) \) by \( (x - 1) \). - Divide \( x \) by \( x \) to get \( 1 \). 5. **Multiply again**: - Multiply \( 1 \) by \( (x - 1) \): \[ 1 \cdot (x - 1) = x - 1. \] 6. **Subtract once more**: - Now subtract: \[ (x + 2) - (x - 1) = 2 + 1 = 3. \] Now, we can summarize our result: \[ \left(x^{2}+2\right) \div(x-1) = x + 1 + \frac{3}{x-1}. \] So, the final answer is: \[ \boxed{x + 1 + \frac{3}{x-1}}. \]