How do you divide #(x^3 - x^2 - 7x - 1) / (x+3) # using polynomial long division?

To divide (x^3 - x^2 - 7x - 1) by (x+3) using polynomial long division, follow these steps:

1. Arrange the dividend (x^3 - x^2 - 7x - 1) and the divisor (x+3) in descending order of powers of x.
2. Divide the first term of the dividend (x^3) by the first term of the divisor (x). The result is x^2.
3. Multiply the divisor (x+3) by the quotient obtained in step 2 (x^2). The result is (x^2)(x+3) = x^3 + 3x^2.
4. Subtract the product obtained in step 3 from the dividend. (x^3 - x^2 - 7x - 1) - (x^3 + 3x^2) = -4x^2 - 7x - 1.
5. Bring down the next term from the dividend, which is -4x^2.
6. Divide the first term of the new dividend (-4x^2) by the first term of the divisor (x). The result is -4x.
7. Multiply the divisor (x+3) by the quotient obtained in step 6 (-4x). The result is (-4x)(x+3) = -4x^2 - 12x.
8. Subtract the product obtained in step 7 from the new dividend. (-4x^2 - 7x - 1) - (-4x^2 - 12x) = 5x - 1.
9. Bring down the next term from the dividend, which is 5x.
10. Divide the first term of the new dividend (5x) by the first term of the divisor (x). The result is 5.
11. Multiply the divisor (x+3) by the quotient obtained in step 10 (5). The result is (5)(x+3) = 5x + 15.
12. Subtract the product obtained in step 11 from the new dividend. (5x - 1) - (5x + 15) = -16.
13. Since there are no more terms in the dividend, the division is complete.
14. The quotient is x^2 - 4x + 5, and the remainder is -16.