Add and/or subtract; simplify completely. List ALL domain restrictions. (You can leave the denominators in factored form.)
3x+2x2+5x−24−7x2+4x−32=
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Let's work through the problem step-by-step:

### Given Expression:
\[
\frac{3x + 2x^2 + 5x - 24}{4 - 7x^2 + 4x - 32}
\]

### Step 1: Combine Like Terms (if necessary)
First, let's simplify the expression in the numerator:

1. **Numerator**: Combine the like terms \(3x\) and \(5x\):
   \[
   3x + 5x = 8x
   \]
   So, the numerator simplifies to:
   \[
   2x^2 + 8x - 24
   \]

2. **Denominator**: Let's simplify the denominator by combining like terms, but in this case, no like terms need combining. Let's also rewrite it in standard form:
   \[
   -7x^2 + 4x + (4 - 32)
   \]
   Simplify the constant terms:
   \[
   4 - 32 = -28
   \]
   So, the denominator simplifies to:
   \[
   -7x^2 + 4x - 28
   \]

### Step 2: Factor the numerator and denominator

1. **Factor the numerator** \(2x^2 + 8x - 24\):
   Factor out the greatest common factor, which is 2:
   \[
   2(x^2 + 4x - 12)
   \]
   Now, factor the quadratic \(x^2 + 4x - 12\):
   \[
   x^2 + 4x - 12 = (x + 6)(x - 2)
   \]
   So, the fully factored numerator is:
   \[
   2(x + 6)(x - 2)
   \]

2. **Factor the denominator** \(-7x^2 + 4x - 28\):
   Factor out the greatest common factor, which is -7:
   \[
   -7(x^2 - \frac{4x}{7} + 4)
   \]
   Notice that this is not easily factorable further without using the quadratic formula, but it does not appear as a simple factorable quadratic. For now, we'll leave it as is:
   \[
   -7(x^2 - \frac{4x}{7} + 4)
   \]

### Step 3: Simplify the Expression

Given the above factors, the expression does not simplify nicely (no cancellation is possible):

\[
\frac{2(x + 6)(x - 2)}{-7(x^2 - \frac{4x}{7} + 4)}
\]

### Step 4: Determine the Domain Restrictions

The domain restrictions occur where the denominator is equal to zero. So, we need to solve:

\[
-7(x^2 - \frac{4x}{7} + 4) = 0
\]

This simplifies to solving:

\[
x^2 - \frac{4x}{7} + 4 = 0
\]

To determine the restrictions, we would solve this quadratic equation using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Where \(a = 1\), \(b = -\frac{4}{7}\), and \(c = 4\). However, solving this leads to complex numbers, which implies that the expression has no real roots, and hence no real domain restrictions from the denominator.

### Final Answer:

Simplified expression:
\[
\boxed{\frac{2(x + 6)(x - 2)}{-7(x^2 - \frac{4x}{7} + 4)}}
\]

Domain restrictions: None for real numbers.

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Would you like further details on any part of this solution or have any additional questions?

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Here are 5 related questions for further exploration:

1. How do you handle the addition or subtraction of fractions with different denominators?
2. Can the domain of a rational expression ever be the entire set of real numbers?
3. How do complex roots of a polynomial affect the domain restrictions of a rational expression?
4. What is the quadratic formula and when is it used?
5. What are the steps to solve a quadratic equation by completing the square?

**Tip:** When simplifying rational expressions, always start by factoring both the numerator and the denominator completely. This can help you identify and cancel common factors efficiently.

Learn how to simplify the rational expression 2(x + 6)(x - 2) / -7(x^2 - 4x/7 + 4) step-by-step. This problem involves factoring the numerator and denominator, and identifying domain restrictions. Suitable for high school students studying algebra and rational expressions.

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