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Let's solve this step by step. The problem involves multiplying two rational expressions and simplifying them, including identifying any restrictions on the variable.

### Problem Setup
The given expression is:

\[
\frac{x^2 + 6x - 27}{x^2 + 2x - 15} \cdot \frac{2x + 10}{x + 4}
\]

### Step 1: Factor Each Polynomial
We need to factor both the numerators and the denominators completely to simplify the expression.

#### Factor the First Numerator: \( x^2 + 6x - 27 \)
This quadratic can be factored as:
\[
x^2 + 6x - 27 = (x + 9)(x - 3)
\]

#### Factor the First Denominator: \( x^2 + 2x - 15 \)
This quadratic can be factored as:
\[
x^2 + 2x - 15 = (x + 5)(x - 3)
\]

#### Factor the Second Numerator: \( 2x + 10 \)
This can be factored by taking out the common factor of 2:
\[
2x + 10 = 2(x + 5)
\]

#### Factor the Second Denominator: \( x + 4 \)
The term \( x + 4 \) is already in its simplest form.

### Step 2: Substitute the Factored Forms
Substitute all these factored forms back into the expression:

\[
\frac{(x + 9)(x - 3)}{(x + 5)(x - 3)} \cdot \frac{2(x + 5)}{x + 4}
\]

### Step 3: Simplify by Cancelling Common Terms
Now we can cancel out terms that appear in both the numerator and the denominator:

1. The \( (x - 3) \) term in the first fraction cancels out.
2. The \( (x + 5) \) term cancels out as well.

This leaves us with:

\[
\frac{(x + 9) \cdot 2}{x + 4} = \frac{2(x + 9)}{x + 4}
\]

### Step 4: State Any Restrictions on the Variable
Restrictions occur when the denominator of any fraction equals zero. So we need to check the denominators in the original expression:

1. \( x^2 + 2x - 15 = 0 \) factors to \( (x + 5)(x - 3) = 0 \), so \( x \neq -5 \) and \( x \neq 3 \).
2. \( x + 4 = 0 \) implies \( x \neq -4 \).

Thus, the restrictions are:
\[
x \neq -5, \quad x \neq 3, \quad x \neq -4
\]

### Final Answer
The simplified expression is:

\[
\frac{2(x + 9)}{x + 4}
\]

with the restrictions \( x \neq -5 \), \( x \neq 3 \), and \( x \neq -4 \).

---

Would you like further details on any of these steps, or do you have questions about another part of the problem?

### Related Questions
1. How do you determine restrictions on a rational expression?
2. Why is it important to factor polynomials when simplifying rational expressions?
3. What are common factoring techniques for quadratic expressions?
4. How do you multiply and divide rational expressions in general?
5. How do you handle restrictions when combining multiple rational expressions?

### Tip
Always check for factors that can cancel out in both the numerator and the denominator, but be careful to first note any restrictions on the variable to avoid dividing by zero.

Multiply: (x^2 + 6x - 27) / (x^2 + 2x - 15) * (2x + 10) / (x + 4) and state any restrictions on the variable.

Learn how to multiply and simplify rational expressions, factoring quadratic expressions and determining restrictions on the variable. This problem is suitable for high school algebra students in grades 9-12.

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