Equation you want to divide is equal to:
12x^3 - 14x^2 + 11x - 20
you are dividing this by:
3x-5
This is very similar to regular long division with some minor adjustmenjts.
First you divide the highest order of the divisor into the highest order of the dividend.
It helps to order both your equations in order of their exponents with the highest exponent first followed by the next highest, etc.
Your equations are already in this order.
Divide 3x into 12x^3
It goes 4x^2 times because 3x * 4x^2 = 12x^3
Put 4x^2 on top and then multiply (3x-5) by 4x^2 and put the result of that under the appropriate ordered term of the dividend.
In this case, the result is 12x^3 -20x^2 and you place it underneath as shown.
Then you SUBTRACT 12x^3 - 20x^2 from 12x^3 - 14x^2.
-14x^2 - (-20x^2) = -14x^2 + 20x^2 = 6x^2.
You are subtracting, but you have to be careful of the signs.
When you subtract a minus, it's the same as adding a plus.
When you add a minus, it's the same as subtracting a plus.
You started off with -14x^2.
You subtracted a -20x^2.
That's the same as adding a +20x^2
-14x^2 + 20x^2 = 6x^2.
You are always subtracting, but you have to be careful of the signs.
Anyway, you bring down the result to get 6x^2 and then you add the next lower order term to it.
That gets you 6x^2 + 11x as shown.
You divide 3x into 6x^2 to get 2x.
You put that on top so now you have 4x^2 + 2x.
You multiply 3x-5 by 2x to get 6x^2 - 10x.
Now you subtract 6x^2 - 10x from 6x^2 + 11x.
11x minus -10x is the same as 11x plus 10x and that equals 21x.
When you subtract a minus, it's the same as adding a plus.
6x^2 - 6x^2 = 0.
You bring down the next lower order term and you have 21x - 20
You divide 3x into 21x to get 7.
You multiply 7 times 3x-5 to get 21x - 35.
You subtract 21x - 35 from 21x - 20 to get 15
-20 minus -35 is the same as -20 plus 35 and the result is 15.
Once again you are subtracting a minus which is equivalent to adding a plus.
You confirm your answer is correct by multiplying the result with the divisor as shown in the picture.
ADDITIONAL INFORMATION
By bringing down the next lower order term, I mean the next lower order term that SHOULD be there, but might not necessarily be there.
You avoid this problem by rewriting your divident is the next lower order term is not there.
Example:
Divide 3x-5 into 15x^5 + 32
The next lower order term is x^4 which is not there, then x^3 which is not there, then x^2 which is not there, then x which is not there.
You write your dividend as follows:
5x^5 + 32 = 15x^5 + 0x^4 + 0x^3 + 0x^2 + 0x + 35
Now the next lower order terms are all there and you can avoid possible confusion further down.
Your first confusion would have been when you divided 3x into 15x^5 to get 5x^4
Then you would multiply 5x^4 by 3x-5 to get 15x^5 - 25x^4
Now you want to subtract 15x^5 - 25x^4 from 15x^5 + 0x^4.
If you did not put that 0x^4 in the dividend to start with, you would have been confused and everything would have been all out of place.