find each quotient, using long division (5x^4+14x^3+9x) / (x^2+3x+1)

To find the quotient using long division, follow these steps:

Step 1: Arrange the dividend and divisor in descending powers of the variable.
The dividend is 5x^4 + 14x^3 + 9x, and the divisor is x^2 + 3x + 1.

Step 2: Divide the first term of the dividend by the first term of the divisor.
Divide 5x^4 by x^2 to get 5x^2.

Step 3: Multiply the divisor by the result from step 2.
Multiply x^2 + 3x + 1 by 5x^2 to get 5x^4 + 15x^3 + 5x^2.

Step 4: Subtract the product obtained in step 3 from the original dividend.
Subtract (5x^4 + 15x^3 + 5x^2) from (5x^4 + 14x^3 + 9x) to get -x^3 + 4x^2 + 9x.

Step 5: Bring down the next term from the dividend.
Bring down -x^3 to the remainder obtained in step 4.

Step 6: Divide the new dividend by the divisor (x^2 + 3x + 1).
Divide -x^3 + 4x^2 + 9x by x^2 to get -x.

Step 7: Multiply the divisor by the result from step 6.
Multiply x^2 + 3x + 1 by -x to get -x^3 - 3x^2 - x.

Step 8: Subtract the product obtained in step 7 from the remainder in step 6.
Subtract (-x^3 - 3x^2 - x) from (-x^3 + 4x^2 + 9x) to get 7x^2 + 10x.

Step 9: Bring down the next term from the dividend.
Bring down 7x^2 to the remainder obtained in step 8.

Step 10: Divide the new dividend by the divisor (x^2 + 3x + 1).
Divide 7x^2 + 10x by x^2 to get 7.

Step 11: Multiply the divisor by the result from step 10.
Multiply x^2 + 3x + 1 by 7 to get 7x^2 + 21x + 7.

Step 12: Subtract the product obtained in step 11 from the remainder in step 10.
Subtract (7x^2 + 21x + 7) from (7x^2 + 10x) to get -11x + 7.

Step 13: There is no more term to bring down from the dividend, and the remainder (-11x + 7) does not have a higher degree than the divisor (x^2 + 3x + 1).

Therefore, the quotient is 5x^2 - x + 7 and the remainder is -11x + 7.