What is the LCD of this problem and the solution?

(x+5)/(x^2+7x+12) + (x^2+8x+15)/(X^2+4x+3)

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Simplify `(x+5)/(x^2+7x+12)+(x^2+8x+15)/(x^2+4x+3)` :

To add rational expressions, like fractions, we need to find a common denominator. The best is the least common multiple of the denominators. The easiest way to find the LCM is to factor:

`x^2+7x+12=(x+3)(x+4)`

`x^2+4x+3=(x+3)(x+1)`

The LCM/LCD has every factor that appears in the factored forms to the highest degree that it appears. So the LCD is (x+3)(x+4)(x+1).

`(x+5)/(x^2+7x+12)+(x^2+8x+15)/(x^2+4x+3)`

`=(x+5)/((x+3)(x+4))+((x+3)(x+5))/((x+3)(x+1))`

`=(x+5)/((x+3)(x+4))*(x+1)/(x+1)+((x+3)(x+5))/((x+3)(x+1))*(x+4)/(x+4)`

`=((x+5)(x+1))/((x+3)(x+4)(x+1))+((x+3)(x+5)(x+4))/((x+3)(x+1)(x+4))`

Now the rational expressions have a common denominator so add the numerators:

`=((x+5)(x+1)+(x+3)(x+5)(x+4))/((x+3)(x+4)(x+1))`

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