(8x^3 + 12x^2 - 6x + 4) ÷ (4x^2 - 2x + 1)

First, we divide the leading term of the numerator by the leading term of the denominator:
8x^3 ÷ 4x^2 = 2x

Next, we multiply the entire denominator by 2x:
2x(4x^2 - 2x + 1) = 8x^3 - 4x^2 + 2x

Subtract this result from the numerator:
8x^3 + 12x^2 - 6x + 4 - (8x^3 - 4x^2 + 2x) = 16x^2 - 8x + 4

Now, we continue the division process by repeating the steps above with the new result 16x^2 - 8x + 4 as the new numerator.

(x^4 + x^2 - 4 + 2x^3) ÷ (x + 2)

First, we divide the leading term of the numerator by the leading term of the denominator:
x^4 ÷ x = x^3

Next, we multiply the entire denominator by x^3:
x^3(x + 2) = x^4 + 2x^3

Subtract this result from the numerator:
x^4 + x^2 - 4 + 2x^3 - (x^4 + 2x^3) = x^2 - 4

Now, we continue the division process by repeating the steps above with the new result x^2 - 4 as the new numerator.