The problem asks us to add two rational expressions and simplify the result: $\frac{6}{x^2 - 11x + 28} + \frac{5}{x^2 - 4x}$

1. Problem Description

The problem asks us to add two rational expressions and simplify the result:

$\frac{6}{x^2 - 11x + 28} + \frac{5}{x^2 - 4x}$

2. Solution Steps

First, factor the denominators:

$x^2 - 11x + 28 = (x - 4)(x - 7)$

$x^2 - 4x = x(x - 4)$

So, the given expression becomes:

$\frac{6}{(x - 4)(x - 7)} + \frac{5}{x(x - 4)}$

To add the two fractions, we need a common denominator. The least common denominator (LCD) is $x(x - 4)(x - 7)$.

Rewrite each fraction with the LCD:

$\frac{6}{(x - 4)(x - 7)} \cdot \frac{x}{x} = \frac{6x}{x(x - 4)(x - 7)}$

$\frac{5}{x(x - 4)} \cdot \frac{(x - 7)}{(x - 7)} = \frac{5(x - 7)}{x(x - 4)(x - 7)}$

Now, add the two fractions:

$\frac{6x}{x(x - 4)(x - 7)} + \frac{5(x - 7)}{x(x - 4)(x - 7)} = \frac{6x + 5(x - 7)}{x(x - 4)(x - 7)}$

Simplify the numerator:

$6x + 5(x - 7) = 6x + 5x - 35 = 11x - 35$

So the expression becomes:

$\frac{11x - 35}{x(x - 4)(x - 7)}$

Expand the denominator:

$x(x - 4)(x - 7) = x(x^2 - 7x - 4x + 28) = x(x^2 - 11x + 28) = x^3 - 11x^2 + 28x$

So the simplified expression is:

$\frac{11x - 35}{x^3 - 11x^2 + 28x}$

3. Final Answer

$\frac{11x - 35}{x(x - 4)(x - 7)}$

or

$\frac{11x - 35}{x^3 - 11x^2 + 28x}$