Q:  (2x + 6x³ + 5) ÷ (2x² + 4)

A: OK, as I tried to do in my previous polynomial long division example, I will try to format this like a “long division” problem the best I can

First, I will re-arrange to get: (6x³ + 2x + 5) ÷ (2x² + 4)

2x² + 4 | 6x³ + 2x + 5

Ok, we start by asking: How many times does 2x² go into 6x³?  The answer: 3x… Which goes about the “long division problem”:

………….. 3x

2x² + 4 | 6x³ + 2x + 5

Now, we multiply 3x by both 2x² and +4.  This then will get carried down in teal like so:

………….. 3x

2x² + 4 | 6x³ + 2x + 5

……………6x³ + 12x

Now, we subtract the teal from the red:

………….. 3x

2x² + 4 | 6x³ + 2x + 5

………….-(6x³ + 12x)

…………………..--10x + 5

Now, since 2x² cannot go into -10x, we are done.

Our answer is 3x + (-10x + 5)/(2x² + 4)

Another way of saying this is: 3x is the quotient and -10x + 5 is the remainder.