simplify the sum.
d^2-7d+12/ d^2-d-6 + d^2+5d+6/ d^2-2d-15
a) 2d^2-5d+24/(d+2) (d-5)
b) 2d^2-13d+24/(d+2) (d-5)
c) 2d^2+13d+24/(d+2) (d-5)
d) 2d^2+5d+24/(d+2) (d-5)
To simplify the sum, we first need to find common factors between the terms in the numerator and denominator.
The numerator is: d^2 - 7d + 12 = (d - 3)(d - 4)
The denominator of the first fraction is: d^2 - d - 6 = (d - 3)(d + 2)
The numerator of the second fraction is: d^2 + 5d + 6 = (d + 2)(d + 3)
The denominator of the second fraction is: d^2 - 2d - 15 = (d - 5)(d + 3)
Now, let's rewrite the original expression:
((d - 3)(d - 4) / ((d - 3)(d + 2)) + ((d + 2)(d + 3)) / ((d - 5)(d + 3))
Next, let's simplify the expression by canceling out common factors:
((d - 4) / (d + 2)) + ((d + 2) / (d - 5))
Now, let's find a common denominator for the fractions and add them together:
[(d - 4)(d - 5) + (d + 2)(d + 2)] / [(d + 2)(d - 5)]
Multiplying out the numerators:
[(d^2 - 9d + 20) + (d^2 + 4d + 4)] / [(d + 2)(d - 5)]
Combining like terms in the numerator:
(2d^2 - 5d + 24) / [(d + 2)(d - 5)]
Therefore, the simplified sum is (a) 2d^2 - 5d + 24 / (d + 2)(d - 5).