Essay. Show all work. Find the quotient

y^4+3y-1
--------
y^2-3

To find the quotient, we will use long division. First, we need to set up the division problem by dividing the leading term of the dividend (y^4) by the leading term of the divisor (y^2).

y^2 + ?

Next, we multiply the divisor (y^2 - 3) by the quotient we just found and subtract the result from the dividend.

y^2 + ?
_______________________
y^2 - 3 | y^4 + 3y - 1

To cancel out the y^4 term, we need to multiply the divisor (y^2 - 3) by y^2.

y^2 + ?
_________________________
y^2 - 3 | y^4 + 3y - 1
- y^4 - 3y^2

Subtracting this result from the dividend:

y^2 + ?
_________________________
y^2 - 3 | y^4 + 3y - 1
- y^4 - 3y^2
_______________
3y^2 + 3y

Now we continue the process. We divide the leading term of what is left (3y^2) by the leading term of the divisor (y^2):

y^2 + 3
_______________________
y^2 - 3 | y^4 + 3y - 1
- y^4 - 3y^2
_______________
3y^2 + 3y
- 3y^2 + 9

Next, we multiply the divisor (y^2 - 3) by the new quotient (3):

y^2 + 3
__________________________
y^2 - 3 | y^4 + 3y - 1
- y^4 - 3y^2
_________________
3y^2 + 3y
- 3y^2 - 9
______________
12

Now we have reached the end of the division process, and the remainder is 12.

Therefore, the quotient is y^2 + 3 and the remainder is 12.

So, the expression

(y^4 + 3y - 1) / (y^2 - 3)

can be simplified as

y^2 + 3 + (12 / (y^2 - 3)).