Perform the operations and simplify the result when possible. $\frac { 5 } { x ^ { 2 } - 3 x - 4 } - \frac { x } { 3 x ^ { 2 } - 10 x - 8 }$
A) $\frac { - x ^ { 2 } + 14 x + 10 } { ( x - 4 ) ( x + 1 ) ( 3 x + 2 ) }$
B) $\frac { x ^ { 2 } + 14 x + 10 } { ( x - 3 ) ( x + 1 ) ( 3 x + 2 ) }$
C) $\frac { - x ^ { 2 } + 14 x + 10 } { ( x - 4 ) ( x + 1 ) \left( 3 x ^ { 2 } - 7 x - 6 \right) }$
D) $\frac { - x ^ { 2 } + 14 x + 10 } { ( x + 1 ) ( 3 x + 2 ) }$
E) $\frac { - x ^ { 2 } + 10 x + 10 } { ( x - 4 ) ( x + 1 ) ( 3 x + 2 ) }$

Question

Quizplus · Accepted Answer

To simplify the given expression, first factor the denominators:1. $x^2 - 3x - 4$ factors into $(x - 4)(x + 1)$.2. $3x^2 - 10x - 8$ can be factored by first factoring out a common factor of 1, resulting in $(3x + 2)(x - 4)$.The least common denominator (LCD) is $(x - 4)(x + 1)(3x + 2)$.Next, adjust the numerators to have the same denominator:1. For the first fraction, multiply the numerator and denominator by $(3x + 2)$ to get $\frac{5(3x + 2)}{(x - 4)(x + 1)(3x + 2)}$.2. For the second fraction, multiply the numerator and denominator by $(x + 1)$ to get $\frac{x(x + 1)}{(x - 4)(x + 1)(3x + 2)}$.Combine the fractions:$\frac{5(3x + 2) - x(x + 1)}{(x - 4)(x + 1)(3x + 2)}$Simplify the numerator:$15x + 10 - x^2 - x$$-x^2 + 14x + 10$Thus, the simplified expression is:$\frac{-x^2 + 14x + 10}{(x - 4)(x + 1)(3x + 2)}$This matches choice A.

Perform the Operations and Simplify the Result When Possible 5x2−3x−4−x3x2−10x−8\frac { 5 } { x ^ { 2 } - 3 x - 4 } - \frac { x } { 3 x ^ { 2 } - 10 x - 8 }x2−3x−45​−3x2−10x−8x​

Perform the Operations and Simplify the Result When Possible $\frac { 5 } { x ^ { 2 } - 3 x - 4 } - \frac { x } { 3 x ^ { 2 } - 10 x - 8 }$