To answer the question, we need to carefully analyze the given expression [tex]\(\left(x^4+3 x^3-2 x^3\right)\left(-5 x^2+x\right)\)[/tex]. We are given the expanded form on the right-hand side of the equation, which is [tex]\(\left(x^4 + 3 x^3 - 2 x^3\right) \left(-5 x^2\right) + \left(x^4 + 3 x^3 - 2 x^3\right) (x)\)[/tex].
Let's go through a step-by-step explanation:
1. Simplifying the First Polynomial:
[tex]\[
x^4 + 3x^3 - 2x^3 = x^4 + (3x^3 - 2x^3) = x^4 + x^3
\][/tex]
2. Rewriting the Expression:
Substitute [tex]\( x^4 + x^3 \)[/tex] back into the expression:
[tex]\[
(x^4 + x^3)(-5x^2 + x)
\][/tex]
3. Applying the Distributive Property:
The original expression has been broken down into two separate products using the distributive property:
[tex]\[
(x^4 + x^3)(-5x^2) + (x^4 + x^3)(x)
\][/tex]
4. Breaking Down the Products:
- For the first product: [tex]\((x^4 + x^3)(-5x^2)\)[/tex]
- For the second product: [tex]\((x^4 + x^3)(x)\)[/tex]
5. Detailed Expansion:
- Expanding [tex]\((x^4 + x^3)(-5x^2)\)[/tex]:
[tex]\[
x^4(-5x^2) + x^3(-5x^2) = -5x^6 - 5x^5
\][/tex]
- Expanding [tex]\((x^4 + x^3)(x)\)[/tex]:
[tex]\[
x^4(x) + x^3(x) = x^5 + x^4
\][/tex]
6. Combining the Results:
Adding the products together:
[tex]\[
-5x^6 - 5x^5 + x^5 + x^4
\][/tex]
The distributive property is used when you multiply a single expression by each term in a binomial and then add the results together. This demonstrates exactly that process.
Therefore, the correct answer is:
[tex]\[ \boxed{\text{C. The distributive property}} \][/tex]
