Question

add.
\\(\dfrac{9}{4x^2 - x - 5} + \dfrac{3}{4x^2 - 13x + 10}\\)
simplify your answer as much as possible.

Explanation:

Step1: Factor denominators

Factor $4x^2 - x - 5 = (4x - 5)(x + 1)$
Factor $4x^2 - 13x + 10 = (4x - 5)(x - 2)$

Step2: Find common denominator

Common denominator is $(4x - 5)(x + 1)(x - 2)$

Step3: Rewrite fractions

$\frac{9(x - 2)}{(4x - 5)(x + 1)(x - 2)} + \frac{3(x + 1)}{(4x - 5)(x + 1)(x - 2)}$

Step4: Expand numerators

$9(x - 2) = 9x - 18$
$3(x + 1) = 3x + 3$

Step5: Combine numerators

$\frac{(9x - 18) + (3x + 3)}{(4x - 5)(x + 1)(x - 2)} = \frac{12x - 15}{(4x - 5)(x + 1)(x - 2)}$

Step6: Simplify numerator

Factor $12x - 15 = 3(4x - 5)$

Step7: Cancel common factors

$\frac{3(4x - 5)}{(4x - 5)(x + 1)(x - 2)} = \frac{3}{(x + 1)(x - 2)}$

Step8: Expand final denominator

$(x + 1)(x - 2) = x^2 - x - 2$

Answer:

$\frac{3}{x^2 - x - 2}$ (or $\frac{3}{(x+1)(x-2)}$)