Below is the addition of these two fractions, but I don't know that it's worth calling this "simplifying."
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The problem is you may have meant this:
(2b - 5) / (b^2 - 2b - 15) + (3b / (b^2 - b - 30)) + ((b^2 + 8b + 12) / (b + 3) )
(2b - 5) / (b - 5)(b + 3) + (3b) / (b - 6)(b + 5) + (b + 2)(b + 6) / (b + 3)
Left denominator needs (b - 6)(b + 5)
(2b - 5)(b - 6)(b + 5) / (b - 5)(b + 3)(b - 6)(b + 5) + (3b) / (b - 6)(b + 5) + (b + 2)(b + 6) / (b + 3)
Middle denominator needs (b - 5)(b + 3)
(2b - 5)(b - 6)(b + 5) / (b - 5)(b + 3)(b - 6)(b + 5) + (3b)(b - 5)(b + 3) / (b - 6)(b + 5)(b - 5)(b + 3) + (b + 2)(b + 6) / (b + 3)
Right denominator needs (b - 6)(b + 5)(b - 5)
(2b - 5)(b - 6)(b + 5) / (b - 5)(b + 3)(b - 6)(b + 5) + (3b)(b - 5)(b + 3) / (b - 6)(b + 5)(b - 5)(b + 3) + (b + 2)(b + 6)(b - 6)(b + 5)(b - 5) / (b + 3)(b - 6)(b + 5)(b - 5)
The denominators are all the same now, so we can do this:
( (2b - 5)(b - 6)(b + 5) + (3b)(b - 5)(b + 3) + (b + 2)(b + 6)(b - 6)(b + 5)(b - 5) ) / (b + 3)(b - 6)(b + 5)(b - 5)
( (2b - 5)(b^2 - b - 30) + (3b)(b^2 - 2b - 15) + (b + 2)(b^2 - 36)(b^2 - 25) ) / (b + 3)(b - 6)(b + 5)(b - 5)
( (b^3 - 2b^2 - 60b - 5b^2 + 5b + 150 + 3b^3 - 6b^2 - 45b + (b + 2)(b^4 - 61b^2 + 900) ) / (b + 3)(b - 6)(b + 5)(b - 5)
(b^3 - 2b^2 - 60b - 5b^2 + 5b + 150 + 3b^3 - 6b^2 - 45b + b^5 - 61b^3 + 900b + 2b^4 - 122b^2 + 1800 ) / (b + 3)(b - 6)(b + 5)(b - 5)
( b^5 + 2b^4 - 57b^3 - 135b^2 + 800b + 1950) / (b + 3)(b - 6)(b + 5)(b - 5)
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But you wrote this:
(2b - 5) / (b^2 - 2b - 15) + (3b / (b^2 - b - 30)) * ((b^2 + 8b + 12) / (b + 3) )
Then:
(2b - 5) / (b - 5)(b + 3) + ( 3b / (b-6)(b+5) ) * ( (b+2)(b+6) / (b+3) )
(2b - 5) / (b - 5)(b + 3) + 3b(b+2)(b+6) / (b-6)(b+5)(b+3)
The denominator on the left needs (b-6)(b+5) so:
(2b - 5)(b - 6)(b + 5) / (b - 5)(b + 3)(b - 6)(b + 5) + 3b(b + 2)(b + 6) / (b - 6)(b + 5)(b + 3)
The denominator on the right needs (b-5) so:
(2b - 5)(b - 6)(b + 5) / (b - 5)(b + 3)(b - 6)(b + 5) + 3b(b + 2)(b + 6)(b - 5) / (b - 6)(b + 5)(b + 3)(b - 5)
Now the denominators are the same, so we can add the numerators together:
( (2b - 5)(b - 6)(b + 5) + 3b(b + 2)(b + 6)(b - 5) ) / (b - 6)(b + 5)(b + 3)(b - 5)
( (2b - 5)(b^2 - b - 30) + 3b(b + 2)(b^2 + b - 30) ) / (b - 6)(b + 5)(b + 3)(b - 5)
( 2b^3 - 2b^2 - 60b - 5b^2 + 5b + 150 + 3b(b^3 + b^2 - 30b + 2b^2 + 2b - 60) ) / (b - 6)(b + 5)(b + 3)(b - 5)
( 2b^3 - 2b^2 - 60b - 5b^2 + 5b + 150 + 3b^4 + 3b^3 - 90b^2 + 6b^3 + 6b^2 - 180b ) / (b - 6)(b + 5)(b + 3)(b - 5)
( 3b^4 + 11b^3 - 91b^2 - 235b + 150 ) / (b - 6)(b + 5)(b + 3)(b - 5)
The top does not have any rational roots, so there's nothing to cancel with the bottom.
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It's a lot of figuring, so it's entirely possible I made a mistake. Feel free to let me know if I messed up.