Algebraic Expression Factorization
The problem in the image is asking to factorize the given algebraic expression. Let's do this step-by-step.
First, we have the expression:
(x + 1) / (x - 2) + (x - 1) / (x^2 - 4)
The second denominator, x^2 - 4, is a difference of squares and can be factored as:
x^2 - 4 = (x + 2)(x - 2)
Now let's rewrite the expression with this factored form:
(x + 1) / (x - 2) + (x - 1) / [(x + 2)(x - 2)]
To combine these fractions, they need to have a common denominator. The least common denominator is (x + 2)(x - 2). So, we multiply the numerator and denominator of the first fraction by (x + 2) to have the same denominator for both terms:
[(x + 1)(x + 2)] / [(x - 2)(x + 2)] + (x - 1) / [(x + 2)(x - 2)]
Expanding the numerator of the first term, we get:
(x^2 + 3x + 2) / [(x - 2)(x + 2)] + (x - 1) / [(x + 2)(x - 2)]
Now that we have a common denominator, we can combine the numerators:
(x^2 + 3x + 2 + x - 1) / [(x + 2)(x - 2)]
Combining like terms in the numerator:
(x^2 + 4x + 1) / [(x + 2)(x - 2)]
This is the final factorized form of the given expression. No further simplification is possible unless we further factor the numerator, which in this case, does not factor nicely over the integers or rationals.
