(x)/((x+5)(x+6))-(5)/((x+5)(x+4))
To simplify the given expression, let's find a common denominator for both fractions.
The denominators of the fractions are (x+5)(x+6) and (x+5)(x+4).
The least common denominator (LCD) is the product of all the unique factors from both denominators.
In this case, (x+5) is a common factor in both denominators, and (x+6) and (x+4) are unique factors.
So, to find the LCD, we multiply all the unique factors: (x+5)(x+6)(x+4).
Now, let's rewrite the fractions with the LCD as the common denominator:
(x) / ((x+5)(x+6)) - (5) / ((x+5)(x+4))
Multiplying the numerators and denominators of each fraction by any factors they are missing will not change the value of the fractions, so we can do that to get the common denominator.
For the first fraction, we multiply the numerator and denominator by (x+4):
(x) * (x+4) / ((x+5)(x+6)(x+4))
For the second fraction, we multiply the numerator and denominator by (x+6):
(5) * (x+6) / ((x+5)(x+6)(x+4))
Now, the expression becomes:
(x(x+4) - 5(x+6)) / ((x+5)(x+6)(x+4))
Simplifying the numerator:
(x^2 + 4x - 5x - 30) / ((x+5)(x+6)(x+4))
(x^2 - x - 30) / ((x+5)(x+6)(x+4))
The expression is now simplified.