1/(x+2)(x+3)-1/(x_4)(x_5)
1/(x+2)(x+3)_1/(x_4)(x_5)
(x+a)(x+b)_ab=(x+c)(x_c)+c(a+c)
If you mean
1/((x+2)(x+3)) - 1/((x-4)(x-5))
that would be
((x-4)(x-5) - (x+2)(x+3)) / ((x+2)(x+3)(x-4)(x-5))
= (14-14x)/((x+2)(x+3)(x-4)(x-5))
= (14-14x)/(x^4-4x^3-19x^2+46x+120)
Now sure what you want to do with the other. And try using "-" instead of "_" for minus!
To simplify the given expression:
1. Start by finding the common denominator for both terms in the expression. In this case, we can see that the denominators are a product of two binomials: (x+2)(x+3) and (x-4)(x-5).
2. Next, find the least common multiple (LCM) of the two denominators. To do this, factor each denominator completely:
(x+2)(x+3) = (x+2)(x-4)
(x-4)(x-5)
When we factor both denominators, we observe that the polynomial (x-4) appears in both factors. Hence, the LCM is (x+2)(x-4)(x-5).
3. Now, we can express each fraction with the common denominator:
(1/(x+2)(x+3)) - (1/(x-4)(x-5))
= [1 * (x-4)(x-5)]/[(x+2)(x+3)(x-4)(x-5)] - [1 * (x+2)(x+3)]/[(x+2)(x+3)(x-4)(x-5)]
4. Combine the fractions with the common denominator:
= [(x-4)(x-5) - (x+2)(x+3)]/[(x+2)(x+3)(x-4)(x-5)]
5. Expand and simplify the expression in the numerator:
= [x^2 - 9x + 20 - (x^2 + 5x + 6)]/[(x+2)(x+3)(x-4)(x-5)]
= [x^2 - 9x + 20 - x^2 - 5x - 6]/[(x+2)(x+3)(x-4)(x-5)]
= [-14x + 14]/[(x+2)(x+3)(x-4)(x-5)]
Therefore, the simplified expression is (-14x + 14)/[(x+2)(x+3)(x-4)(x-5)].