Perform the following operation and express in simplest form.

start fraction, 5, x, minus, 20, divided by, x, squared, minus, 12, x, plus, 32, end fraction, plus, start fraction, 7, x, plus, 21, divided by, x, squared, plus, 3, x, end fraction 
x 
2
 −12x+32
5x−20
​
 + 
x 
2
 +3x
7x+21
​
 


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.

Perform the following operation and express in simplest form: \( \frac{5x - 20}{x^2 - 12x + 32} + \frac{7x + 21}{x^2 + 3x} \)

Learn how to simplify the rational expression \( \frac{5x - 20}{x^2 - 12x + 32} + \frac{7x + 21}{x^2 + 3x} \) by factoring the denominators and adding the fractions. The solution covers factoring techniques, finding the least common denominator (LCD), and simplifying rational expressions. Suitable for students in Grades 9-12.

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