# Simplify 2/(y^2-3y+2)+5/(y^2-1)

2y2-3y+2+5y2-1
Simplify each term.
Factor y2-3y+2 using the AC method.
Consider the form x2+bx+c. Find a pair of integers whose product is c and whose sum is b. In this case, whose product is 2 and whose sum is -3.
-2,-1
Write the factored form using these integers.
2(y-2)(y-1)+5y2-1
2(y-2)(y-1)+5y2-1
Simplify the denominator.
Rewrite 1 as 12.
2(y-2)(y-1)+5y2-12
Since both terms are perfect squares, factor using the difference of squares formula, a2-b2=(a+b)(a-b) where a=y and b=1.
2(y-2)(y-1)+5(y+1)(y-1)
2(y-2)(y-1)+5(y+1)(y-1)
2(y-2)(y-1)+5(y+1)(y-1)
To write 2(y-2)(y-1) as a fraction with a common denominator, multiply by y+1y+1.
2(y-2)(y-1)⋅y+1y+1+5(y+1)(y-1)
To write 5(y+1)(y-1) as a fraction with a common denominator, multiply by y-2y-2.
2(y-2)(y-1)⋅y+1y+1+5(y+1)(y-1)⋅y-2y-2
Write each expression with a common denominator of (y-2)(y-1)(y+1), by multiplying each by an appropriate factor of 1.
Multiply 2(y-2)(y-1) and y+1y+1.
2(y+1)(y-2)(y-1)(y+1)+5(y+1)(y-1)⋅y-2y-2
Multiply 5(y+1)(y-1) and y-2y-2.
2(y+1)(y-2)(y-1)(y+1)+5(y-2)(y+1)(y-1)(y-2)
Reorder the factors of (y-2)(y-1)(y+1).
2(y+1)(y+1)(y-1)(y-2)+5(y-2)(y+1)(y-1)(y-2)
2(y+1)(y+1)(y-1)(y-2)+5(y-2)(y+1)(y-1)(y-2)
Combine the numerators over the common denominator.
2(y+1)+5(y-2)(y+1)(y-1)(y-2)
Simplify the numerator.
Apply the distributive property.
2y+2⋅1+5(y-2)(y+1)(y-1)(y-2)
Multiply 2 by 1.
2y+2+5(y-2)(y+1)(y-1)(y-2)
Apply the distributive property.
2y+2+5y+5⋅-2(y+1)(y-1)(y-2)
Multiply 5 by -2.
2y+2+5y-10(y+1)(y-1)(y-2)