https://file.aiquickdraw.com/mathbot/file/17259850839843owgv3ut.jpg

The expression in the image is:

\[
\sqrt{\frac{(3x - 3)(2x^2 + 14)}{(3x^2 + 4x + 15)}}
\]

Let's simplify and analyze the expression step by step:

### Step 1: Simplify the numerator
The numerator is:
\[
(3x - 3)(2x^2 + 14)
\]
You can factor out the constants:
\[
3(x - 1)(2(x^2 + 7)) = 6(x - 1)(x^2 + 7)
\]

### Step 2: Review the denominator
The denominator is:
\[
3x^2 + 4x + 15
\]
This quadratic expression cannot be easily factored, so it remains as is.

### Step 3: Simplify the whole expression
Now, substitute the simplified numerator:
\[
\sqrt{\frac{6(x - 1)(x^2 + 7)}{3x^2 + 4x + 15}}
\]

Since no further simplification is possible between the numerator and denominator, this is the simplest form of the expression.

Would you like further details on this simplification or another breakdown? Here are five related questions:

1. How do you factor more complex quadratic expressions like the denominator?
2. What methods are used to simplify square roots of rational expressions?
3. How do you identify when a polynomial can be factored?
4. What steps would you take to graph this expression?
5. How does multiplying binomials affect the final result in such expressions?

**Tip:** When simplifying, always check if you can factor out constants or common terms early to make the process easier.

Simplify the expression \( \sqrt{\frac{(3x - 3)(2x^2 + 14)}{3x^2 + 4x + 15}} \).

Learn how to simplify the expression \( \sqrt{\frac{(3x - 3)(2x^2 + 14)}{3x^2 + 4x + 15}} \) with a detailed step-by-step solution. This problem involves key algebraic concepts, including simplifying expressions and factoring quadratic equations. Suitable for students in Grades 9-12.

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