https://file.aiquickdraw.com/mathbot/file/1726319202130fxwt90r8.jpg

To determine the degree of the given polynomial, let's analyze the expression step by step based on the image:

The polynomial is:

\[
( x^2 - 5x^2 + 10x - 5 ) \cdot 4 \cdot (4x - 3)^3 \cdot (x^2 + 4x)
\]

### Step-by-step analysis:

1. **First Polynomial:**
   \[
   (x^2 - 5x^2 + 10x - 5) 
   \]
   Simplifying this:
   \[
   x^2 - 5x^2 = -4x^2, \quad \text{so it becomes} \quad -4x^2 + 10x - 5
   \]
   The highest degree term is \(-4x^2\), which has degree 2.

2. **Next Factors:**
   \[
   4 \quad \text{is a constant, so it doesn't affect the degree}.
   \]

3. **Third Polynomial:**
   \[
   (4x - 3)^3
   \]
   When expanded, the degree of \((4x - 3)^3\) is \(3 \times \text{degree of } (4x) = 3 \times 1 = 3\).

4. **Fourth Polynomial:**
   \[
   (x^2 + 4x)
   \]
   The highest degree term is \(x^2\), so its degree is 2.

### Total Degree:

The degree of a product of polynomials is the sum of the degrees of the individual terms.

- Degree of \((-4x^2 + 10x - 5)\) = 2
- Degree of \((4x - 3)^3\) = 3
- Degree of \((x^2 + 4x)\) = 2

Adding these together:

\[
2 + 3 + 2 = 7
\]

Thus, the degree of the polynomial is **7**.

However, none of the answer choices seem to match this. Could it be that the answer choices pertain to a different calculation, or is this expression correct? Let me know if you would like me to explore further or clarify anything!

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Would you like additional details or have any questions? Here are some related questions you might find useful:

1. What is the definition of a polynomial's degree?
2. How do you expand a binomial raised to a power like \((4x - 3)^3\)?
3. Can the degree of a product of two polynomials ever be less than either of the individual polynomials?
4. How does factoring affect the degree of a polynomial?
5. Why don't constants (like 4) affect the degree of a polynomial?

**Tip:** The degree of a polynomial is always determined by the term with the highest total power of \(x\) after simplification.

Find the degree of the polynomial: (x^2 - 5x^2 + 10x - 5) * 4 * (4x - 3)^3 * (x^2 + 4x)

Explore how to find the degree of the polynomial expression: (x^2 - 5x^2 + 10x - 5) * 4 * (4x - 3)^3 * (x^2 + 4x). Learn the step-by-step process to simplify and calculate the degree, involving exponents and polynomials, suitable for high school students.

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