How do I rearrange terms in this polynomial to take the square root?

Question

I was given the following problem:

$$\sqrt{6ab^2c - 4a^2bc+a^2b^2+4a^2c^2+9b^2c^2-12abc}$$

While I know how to find the square root of a polynomial, I'm not certain as to how I would arrange the terms in this polynomial so that I could obtain the square root. My book doesn't give me enough details as to how I should arrange polynomials so that I can do the division easily. I tried to do this:

$$6ab^2c - 4a^2bc + a^2b^2c^0 + 4a^2b^0c^2 + 9a^0b^2c^2 - 12abc$$

Basically, I put in redundant terms so I could maybe see the bigger picture. However, it didn't give me anything fruitful. If anyone can explain how to rearrange the terms in this equation, that would be fantastic.

Answer 1

Hint: What is $(ab + 3bc - 2ac)^2$?

Answer 2

Alright, so, right after posting this question, I actually figured it out (why do I do this to myself?). Basically, I was overthinking it: I should've just rearranged the problem like this:

$$6ab^2c−4a^2bc+a^2b^2+4a^2c^2+9b^2c^2−12abc^2 = 4a^2c^2-4a^2bc+a^2b^2+6ab^2c-12abc^2+9b^2c^2$$

In essence, I rearranged the problem in descending powers of $a$ and then obtained $2ac - ab - 3bc$ as my result.

Answer 3

$$\sqrt{6ab^2c - 4a^2bc+a^2b^2+4a^2c^2+9b^2c^2-12abc}=\sqrt { (ab-2ac+3bc)^2 }= | ab-2ac+3bc|$$