Factor out 8x^3y^6 + 27

One factor is 2xy^2+3. (The sum of the cube roots of the first and last terms). Use long division to compute
(8x^3y^6 + 27)/(2xy^2 + 3) to get the other factor

To factor out the expression 8x^3y^6 + 27, we can use the sum of cubes formula, which states that a^3 + b^3 can be factored as (a + b)(a^2 - ab + b^2).

In this case, the cube root of 8x^3y^6 is 2xy^2, and the cube root of 27 is 3. So we can express the given expression as (2xy^2 + 3)(a^2 - ab + b^2).

To find the other factor, we can use long division. Divide 8x^3y^6 + 27 by 2xy^2 + 3:

_________
2xy^2 + 3 | 8x^3y^6 + 0x^2y^4 + 0xy^2 + 27

To begin the long division, ask yourself, "What do I multiply 2xy^2 by to get 8x^3y^6?" The answer is 4xy^4. Write this above the division line, in the position where x^3y^6 would be:

4xy^4
___________
2xy^2 + 3 | 8x^3y^6 + 0x^2y^4 + 0xy^2 + 27

Next, multiply 2xy^2 by 4xy^4 and write the result under the 0x^2y^4 term:

4xy^4
___________
2xy^2 + 3 | 8x^3y^6 + 0x^2y^4 + 0xy^2 + 27
- 8x^3y^6 - 12xy^6

Now, subtract the result from the previous step from the first term of the dividend. Write the result below the line:

4xy^4
___________
2xy^2 + 3 | 8x^3y^6 + 0x^2y^4 + 0xy^2 + 27
- 8x^3y^6 - 12xy^6
___________________
12xy^6 + 0x^2y^4 + 0xy^2 + 27

Now, bring down the next term, which is 0xy^2:

4xy^4
___________
2xy^2 + 3 | 8x^3y^6 + 0x^2y^4 + 0xy^2 + 27
- 8x^3y^6 - 12xy^6
___________________
12xy^6 + 0x^2y^4 + 0xy^2 + 27
- 12xy^6 - 18xy^4

Repeat the process of multiplying and subtracting until there are no more terms left:

4xy^4 - 12xy^4
______________________
2xy^2 + 3 | 8x^3y^6 + 0x^2y^4 + 0xy^2 + 27
- 8x^3y^6 - 12xy^6
____________________
12xy^6 + 0x^2y^4 + 0xy^2 + 27
- 12xy^6 - 18xy^4
_________________
18xy^4 + 0xy^2 + 27
- 18xy^4 - 27x^2y^2
__________________
27x^2y^2 + 0xy^2 + 27

At this point, we can see that the remaining term is 27x^2y^2 + 0xy^2 + 27. This is the remainder, and since there are no more terms left, the division is complete.

The final result of the division is (2xy^2 + 3)(4xy^4 - 12xy^2 + 18y^4 + 27x^2y^2 + 0xy^2 + 27). However, we can simplify this result by combining like terms:

(2xy^2 + 3)(4xy^4 - 12xy^2 + 27x^2y^2 + 18y^4 + 27)

So, the fully simplified factored form of 8x^3y^6 + 27 is (2xy^2 + 3)(4xy^4 - 12xy^2 + 27x^2y^2 + 18y^4 + 27).