Rewrite as equivalent rational expressions with denominator 3(*x*−1)(*x*+5)(*x*+9):

63*x*2+12*x*−15,5*x*3*x*2+24*x*−27.

To rewrite the expressions with the denominator \( 3(x - 1)(x + 5)(x + 9) \), we need to express each term with this common denominator.

### Given expressions:
1. \( 63x^2 + 12x - 15 \)
2. \( 5x^3 + 3x^2 + 24x - 27 \)

We will rewrite both expressions with the denominator \( 3(x - 1)(x + 5)(x + 9) \).

---

### Expression 1: \( 63x^2 + 12x - 15 \)

1. Multiply both the numerator and the denominator of this expression by the required factors \( (x - 1)(x + 5)(x + 9) \), assuming the denominator is currently 1:
   \[
   \frac{63x^2 + 12x - 15}{1} = \frac{(63x^2 + 12x - 15) \cdot 3(x - 1)(x + 5)(x + 9)}{3(x - 1)(x + 5)(x + 9)}
   \]

2. The rewritten form is:
   \[
   \frac{(63x^2 + 12x - 15) \cdot 3(x - 1)(x + 5)(x + 9)}{3(x - 1)(x + 5)(x + 9)}
   \]

---

### Expression 2: \( 5x^3 + 3x^2 + 24x - 27 \)

1. Similarly, multiply the numerator and denominator by the necessary factors to get the denominator \( 3(x - 1)(x + 5)(x + 9) \):
   \[
   \frac{5x^3 + 3x^2 + 24x - 27}{1} = \frac{(5x^3 + 3x^2 + 24x - 27) \cdot (x - 1)(x + 5)(x + 9)}{3(x - 1)(x + 5)(x + 9)}
   \]

2. The rewritten form is:
   \[
   \frac{(5x^3 + 3x^2 + 24x - 27) \cdot (x - 1)(x + 5)(x + 9)}{3(x - 1)(x + 5)(x + 9)}
   \]

---

### Final Rational Expressions:

1. \[
   \frac{(63x^2 + 12x - 15) \cdot 3(x - 1)(x + 5)(x + 9)}{3(x - 1)(x + 5)(x + 9)}
   \]
   
2. \[
   \frac{(5x^3 + 3x^2 + 24x - 27) \cdot (x - 1)(x + 5)(x + 9)}{3(x - 1)(x + 5)(x + 9)}
   \]

Would you like me to expand or simplify these expressions further? Let me know!

Here are 5 related questions:
1. How do you factor quadratic expressions like \( 63x^2 + 12x - 15 \)?
2. What are rational expressions, and why do we use them?
3. How do you determine the common denominator for multiple rational expressions?
4. Can these expressions be simplified further?
5. How do you add or subtract rational expressions with different denominators?

**Tip**: When rewriting expressions with common denominators, always ensure that the numerator is multiplied by all the factors of the new denominator.

Rewrite as equivalent rational expressions with denominator 3(x−1)(x+5)(x+9): 63x^2 + 12x − 15, 5x^3 + 3x^2 + 24x − 27.

Learn how to rewrite the rational expressions 63x^2 + 12x − 15 and 5x^3 + 3x^2 + 24x − 27 with a common denominator of 3(x−1)(x+5)(x+9). This step-by-step guide covers key algebraic concepts such as polynomial multiplication and rational expressions. Suitable for students in Grades 9-11.

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