How do you simplify # 3/(y^2-3y+2 ) + 5/(y^2-1)#?

To simplify the expression 3/(y^2-3y+2) + 5/(y^2-1), we need to find a common denominator and combine the fractions. The common denominator is (y^2-3y+2)(y^2-1).

To add the fractions, we multiply the numerator and denominator of each fraction by the missing factors in the other denominator.

For the first fraction, we multiply the numerator and denominator by (y^2-1):

3/(y^2-3y+2) * (y^2-1)/(y^2-1) = 3(y^2-1)/[(y^2-3y+2)(y^2-1)] = 3(y^2-1)/(y^4-4y^3+5y^2-4y+2)

For the second fraction, we multiply the numerator and denominator by (y^2-3y+2):

5/(y^2-1) * (y^2-3y+2)/(y^2-3y+2) = 5(y^2-3y+2)/[(y^2-3y+2)(y^2-1)] = 5(y^2-3y+2)/(y^4-4y^3+5y^2-4y+2)

Now, we can combine the fractions:

3(y^2-1)/(y^4-4y^3+5y^2-4y+2) + 5(y^2-3y+2)/(y^4-4y^3+5y^2-4y+2) = (3y^2-3+5y^2-15y+10)/(y^4-4y^3+5y^2-4y+2)

Combining like terms in the numerator:

(8y^2-15y+7)/(y^4-4y^3+5y^2-4y+2)

Therefore, the simplified expression is (8y^2-15y+7)/(y^4-4y^3+5y^2-4y+2).