Rewrite as equivalent rational expressions with denominator  
(
5
z
−
4
)
(
z
−
3
)
(
z
+
5
)
 :
5
5
z
2
−
19
z
+
12
,
10
z
5
z
2
+
21
z
−
20


We are asked to rewrite two given rational expressions with the same denominator, which involves finding a common denominator for the expressions. Let's go through the process step by step.

### Given Expressions:
1. \(\frac{5}{5z^2 - 19z + 12}\)
2. \(\frac{10z}{5z^2 + 21z - 20}\)

### Step 1: Factor the Denominators

#### First denominator: \(5z^2 - 19z + 12\)
We need to factor this quadratic expression.

- Multiply \(5 \times 12 = 60\).
- Find two numbers that multiply to 60 and add to \(-19\). These numbers are \(-15\) and \(-4\).
- Rewrite the middle term: 
  \[
  5z^2 - 15z - 4z + 12
  \]
- Factor by grouping:
  \[
  5z(z - 3) - 4(z - 3) = (5z - 4)(z - 3)
  \]
Thus, the first denominator is factored as:
\[
5z^2 - 19z + 12 = (5z - 4)(z - 3)
\]

#### Second denominator: \(5z^2 + 21z - 20\)
Similarly, we factor this quadratic expression.

- Multiply \(5 \times -20 = -100\).
- Find two numbers that multiply to \(-100\) and add to \(21\). These numbers are \(25\) and \(-4\).
- Rewrite the middle term:
  \[
  5z^2 + 25z - 4z - 20
  \]
- Factor by grouping:
  \[
  5z(z + 5) - 4(z + 5) = (5z - 4)(z + 5)
  \]
Thus, the second denominator is factored as:
\[
5z^2 + 21z - 20 = (5z - 4)(z + 5)
\]

### Step 2: Find the Least Common Denominator (LCD)
The factored denominators are:
1. \((5z - 4)(z - 3)\)
2. \((5z - 4)(z + 5)\)

The least common denominator (LCD) must include all factors that appear in any denominator. Therefore, the LCD is:
\[
\text{LCD} = (5z - 4)(z - 3)(z + 5)
\]

### Step 3: Rewrite Each Expression with the Common Denominator

#### First expression: \(\frac{5}{(5z - 4)(z - 3)}\)
To have the common denominator, we need to multiply the numerator and denominator by \((z + 5)\):
\[
\frac{5}{(5z - 4)(z - 3)} \times \frac{(z + 5)}{(z + 5)} = \frac{5(z + 5)}{(5z - 4)(z - 3)(z + 5)}
\]
So the first expression becomes:
\[
\frac{5(z + 5)}{(5z - 4)(z - 3)(z + 5)}
\]

#### Second expression: \(\frac{10z}{(5z - 4)(z + 5)}\)
To have the common denominator, we need to multiply the numerator and denominator by \((z - 3)\):
\[
\frac{10z}{(5z - 4)(z + 5)} \times \frac{(z - 3)}{(z - 3)} = \frac{10z(z - 3)}{(5z - 4)(z - 3)(z + 5)}
\]
So the second expression becomes:
\[
\frac{10z(z - 3)}{(5z - 4)(z - 3)(z + 5)}
\]

### Final Answer:
The two rational expressions with the same denominator are:
\[
\frac{5(z + 5)}{(5z - 4)(z - 3)(z + 5)} \quad \text{and} \quad \frac{10z(z - 3)}{(5z - 4)(z - 3)(z + 5)}
\]

Would you like more details or explanations on any step?

---

### Related Questions:
1. How do you factor more complex quadratics?
2. What is the process for finding the least common denominator for rational expressions?
3. How do you simplify rational expressions after obtaining a common denominator?
4. What is the significance of factoring in solving rational expressions?
5. How do you check if your factorization of a quadratic expression is correct?

**Tip:** Always double-check your factorization by multiplying the factors to ensure they produce the original quadratic expression.

Rewrite as equivalent rational expressions with denominator (5z - 4)(z - 3)(z + 5): 5/(5z^2 - 19z + 12), 10z/(5z^2 + 21z - 20).

Learn how to rewrite rational expressions with quadratic denominators, factoring each expression and finding the least common denominator (LCD). This step-by-step solution covers factoring and using the LCD for rewriting rational expressions. Suitable for students in Grades 9-12.

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