I will attempt this on paper since it's too difficult for me to write here (and the text here does not allow large spacings between characters)

The first thing we have to do in this question is to expand all the brackets to express our dividend (the expression we are trying to divide, like a cake) and our divisor (the expression we will be using as a "knife" to divide the cake).

[Or as a simpler illustration using primary school / lower grade maths, when we divide 7 by 2, our dividend is 7 and our divisor is 2]

We will get the dividend as 81x⁴ - 72x² + 16 and the divisor as 9x² - 6x + 1. For the purposes of long division, we write 81x⁴ - 72x² + 16 as 81x⁴ + 0x³ - 72x² + 0x + 16. Not a must, but it helps to a degree.

Now, let's perform the long division.

For regular numbers, we usually divide bigger numbers by smaller numbers. The other way is of course possible, but this will not be a point of discussion for this question.

How do you divide a number, say 6294, by 3? What are the steps you took to divide this? How about 6294 by 16? Do you still recall?

The approach for the algebraic long division is largely similar, but here the "larger number" is basically an expression whose highest power of x (we call it the degree of the expression or in this case the degree of the polynomial) is bigger than another expression.

Long division can still be carried out if the degrees of the two expressions are equal, but it is not possible if we are trying to divide a polynomial of a lower degree by another polynomial of a higher degree.

We are trying to divide 81x⁴ + 0x³ - 72x² + 0x + 16 by 9x² - 6x + 1.

Our dividend is 81x⁴ + 0x³ - 72x² + 0x + 16 .
Our divisor is 9x² - 6x + 1.

The degree of the dividend is more than (or at least equal to) the degree of the divisor. So, long division can proceed.

Take my example just now. When dividing 6294 by 3, what do we do first? We divide 6 by 3 to get 2 and then place this 2 on top of the 6. Recall?

The idea here is somewhat similar, but with a twist. Currently the term with the highest power of x in the dividend is 81x⁴ while the term with the highest power of x in the divisor is 9x².

We divide 81x⁴ by 9x² to get 9x² (just as we divide 6 by 3 to get 2). We place this 9x² on top of the 81x⁴, analogous to the numbers case. That's the first thing which I wrote in the quotient (the other terms in the quotient come later)

The slight difference here is that we need to multiply the 9x² by the entire divisor 9x² - 6x + 1 and write its result directly one column below the dividend. We then subtract this new column from the dividend to get our subsequent line.

Of course, the 81x⁴ must disappear because that's how we chose 9x² to be written on top (when dividing 6294 by 3, we write "2" and not "1" or "3" as the starting number for a reason).

But the remaining terms are of interest. Upon subtracting the two columns, we get 54x³ - 81x². I proceed to bring down the next term, "0x", from the dividend (similar to the numbers case).

From here, we repeat a similar process. You can think of 54x³ - 81x² + 0x as the new "sub-dividend" for this process.

Note that this long division process continues until we get a "sub-dividend" which is of a lower degree than that of the divisor.

As before, the term with the highest power of x in this "sub-dividend" is 54x³ while the term with the highest power of x in the divisor is 9x².

Dividing 54x³ by 9x² gets us 6x, and we will add this next to the 9x² which we already wrote in the quotient section.

We then multiply this 6x by the entire divisor 9x² - 6x + 1 to get 54x³ - 36x² + 6x which we will write below this "sub-dividend". As before, we subtract the two lines. The term in x³ should disappear (else, our choice would have been incorrect earlier).

We will obtain -45x² - 6x from the subtraction. We then bring down the next term +16 from the quotient, so that our new "sub-dividend" is -45x² - 6x + 16.

This new "sub-dividend" -45x² - 6x + 16 has the same degree as the divisor 9x² - 6x + 1, so we can do a similar process one more time.

Dividing -45x² by 9x² gets us -5 and we include this in our quotient.

Similar to before, we multiply -5 to the divisor 9x² - 6x + 1 to get -45x² + 30x - 5 which we will write below the new "sub-dividend".

We subtract both lines to get -36x + 21.

This newest "sub-dividend" is of a lower degree than the divisor, so the long division process ends here.

Finally, we collect our results.

Our quotient is 9x² + 6x - 5 and our remainder is 21 - 36x.

Putting this in the statement Dividend = Divisor x Quotient + Remainder,

(3x + 2)² (3x - 2)² = (3x - 1)² (9x² + 6x - 5) + 21 - 36x

okay I think I understand much clearer now ! thank you very much for your help <3