Long Division: Finding Quotient and Remainder for x^3 + 4x^2 + 8x + 4 by 4x + 8

To find the quotient and remainder using long division when dividing x^3 + 4x^2 + 8x + 4 by 4x + 8, we follow these steps:

         1/4
   _______________________

4x + 8 | x^3 + 4x^2 + 8x + 4

Step 1: Divide the highest degree term of the dividend (x^3) by the highest degree term of the divisor (4x) to get 1/4x^2.

         1/4
   _______________________

4x + 8 | x^3 + 4x^2 + 8x + 4 - (x^3 + 4x^2)

Step 2: Multiply the divisor (4x + 8) by the quotient term (1/4x^2) and subtract it from the dividend.

         1/4
   _______________________

4x + 8 | x^3 + 4x^2 + 8x + 4 - (x^3 + 4x^2) _______________________ 4x + 4x + 4

Step 3: Repeat steps 1 and 2 with the new dividend (4x + 4x + 4):

         1/4
   _______________________

4x + 8 | x^3 + 4x^2 + 8x + 4 - (x^3 + 4x^2) _______________________ 4x + 4x + 4 - (4x + 4x) _______________________ 4

Step 4: The dividend (4) is of a lower degree than the divisor, so we cannot proceed further.

Therefore, the quotient is 1/4x^2 + 1, and the remainder is 4.