We need to perform the following operation and express the result in its simplest form:

$\frac{5x - 20}{x^2 - 12x + 32} + \frac{7x + 21}{x^2 + 3x}$

Step 1: Factor the Denominators

First denominator: $x^2 - 12x + 32$

We factor the quadratic expression: $x^2 - 12x + 32 = (x - 8)(x - 4)$

So, the first fraction becomes: $\frac{5x - 20}{(x - 8)(x - 4)}$

Since $5x - 20 = 5(x - 4)$, we simplify: $\frac{5(x - 4)}{(x - 8)(x - 4)} = \frac{5}{x - 8}$

Second denominator: $x^2 + 3x$

Factor out the common term: $x^2 + 3x = x(x + 3)$

So, the second fraction becomes: $\frac{7x + 21}{x(x + 3)}$

Since $7x + 21 = 7(x + 3)$, we simplify: $\frac{7(x + 3)}{x(x + 3)} = \frac{7}{x}$

Step 2: Add the Two Fractions

Now, we need to add:

$\frac{5}{x - 8} + \frac{7}{x}$

Since the denominators are different, the least common denominator (LCD) is $x(x - 8)$. Rewrite each fraction with this denominator:

$\frac{5}{x - 8} = \frac{5x}{x(x - 8)}$

$\frac{7}{x} = \frac{7(x - 8)}{x(x - 8)}$

Now add the numerators:

$\frac{5x + 7(x - 8)}{x(x - 8)}$

Distribute:

$\frac{5x + 7x - 56}{x(x - 8)}$

$\frac{12x - 56}{x(x - 8)}$

Factor out 4:

$\frac{4(3x - 14)}{x(x - 8)}$

Final Answer

$\frac{4(3x - 14)}{x(x - 8)}$

Let me know if you need further clarification.

Related Questions:

1. How do you determine the least common denominator (LCD) when adding fractions?
2. What are the steps to factor a quadratic expression?
3. How do you simplify rational expressions?
4. How do you find restrictions in a rational function?
5. What are the key properties of polynomial division?

Tip:

Always factor the denominators first to find a common denominator efficiently when adding rational expressions.