Answer:
(x^2 + 3x + 1) / (x - 1) is x + 4 + 5 / (x - 1).
Explanation:
1. Divide the first term of the numerator, x^2, by the first term of the denominator, x. This gives us x.
2. Multiply the denominator, x - 1, by the quotient we obtained in step 1, which is x. This gives us x * (x - 1) = x^2 - x.
3. Subtract the result from step 2 from the numerator. (x^2 + 3x + 1) - (x^2 - x) = 4x + 1.
4. Bring down the next term from the numerator, which is 4x. We now have 4x + 1 as the new numerator.
5. Divide the first term of the new numerator, 4x, by the first term of the denominator, x. This gives us 4.
6. Multiply the denominator, x - 1, by the quotient we obtained in step 5, which is 4. This gives us 4 * (x - 1) = 4x - 4.
7. Subtract the result from step 6 from the new numerator. (4x + 1) - (4x - 4) = 5.
8. the remainder is 5.