(4y^2-9)/(2y^2+y-3)
always fully factor everything when you're simplifying!
Top is a difference of squares expression, bottom you can simplify by factoring.
TOP:
(?y-?)(?y+?)
First ? of each bracket is square root of first coefficient (4)
Second? of each bracket is square root of second coefficient (9):
(2y-3)(2y+3)
BOTTOM:
(2y^2 +y -3)
3 is negative, therefore one bracket is positive, the other is negative
(?y-?)(?y+?)
First ? of each bracket will be factors of first coefficient (2)
Second ? of each bracket will be factors of last coefficinet (3)
Mix and match until upon multiplying it out, you can get the middle term; 1y;
(2y-3)(y+1)
2y^2 -3 -3y +2y
2y^2 -1y -3
Not quite, so let's switch the signs around:
(2y+3)(y-1)
2y^2 -3 +3y -2y
2y^2 +y -3
Perfect, so our bottom is going to be:
(2y+3)(y-1)
SO NOW WE"VE SIMPLIFIED
(4y^2-9)/(2y^2+y-3)
INTO
(2y+3)(2y-3)/(2y+3)(y-1)
NOTE that the (2y+3)s cancel each other out, leaving you with
1(2y-3)/1(y-1)
(2y-3)/(y-1)
can't factor/simplify further, so this is the final answer. So it's going to be other option 2 or 4.
TO FIND RESTRICTIONS, set each part containing 'y' IN ALL DENOMINATORS WE'VE ENCOUNTERED SO FAR equal to 0. The reason for this; anything multiplied by zero in the denominator makes the equation UNDEFINED, so we need to find out what those values are by setting them equal to 0
denominators we've come across: (2y+3) and (y-1)
(2y+3)=0 and (y-1)=0
2y= -3 and y= 1
y= -3/2 and y= 1
THEREFORE the restrictions we have our y= cannot equal -3/2, 1, THEREFORE THE SECOND OPTION IS CORRECT