Question

(\frac{x + 1}{x^2 - 3x - 28} - \frac{x - 4}{2x - 14})

Explanation:

Step1: Factor denominators

Factor $x^2-3x-28$ as $(x-7)(x+4)$, factor $2x-14$ as $2(x-7)$.
Expression: $\frac{x+1}{(x-7)(x+4)} - \frac{x-4}{2(x-7)}$

Step2: Find common denominator

Common denominator is $2(x-7)(x+4)$. Rewrite fractions:
Expression: $\frac{2(x+1)}{2(x-7)(x+4)} - \frac{(x-4)(x+4)}{2(x-7)(x+4)}$

Step3: Expand numerators

Expand $2(x+1)$ and $(x-4)(x+4)=x^2-16$:
Expression: $\frac{2x+2 - (x^2-16)}{2(x-7)(x+4)}$

Step4: Simplify numerator

Combine like terms in numerator:
Expression: $\frac{2x+2 -x^2 +16}{2(x-7)(x+4)} = \frac{-x^2+2x+18}{2(x-7)(x+4)}$

Step5: Rewrite numerator (optional)

Factor out -1 from numerator:
Expression: $\frac{-(x^2-2x-18)}{2(x-7)(x+4)}$

Answer:

$\boldsymbol{\frac{-x^2+2x+18}{2(x-7)(x+4)}}$ or $\boldsymbol{-\frac{x^2-2x-18}{2(x-7)(x+4)}}$