To solve the problem and determine whether the expression is an example of the distributive property, multiplying two binomials, vertical multiplication, or FOIL, let's analyze the given expression step-by-step.
Given expression:
[tex]\[ (x^4 + 3x^3 - 2x^3)(-5x^2 + x) \][/tex]
First, let's simplify inside the parentheses on the left-hand side:
[tex]\[ x^4 + 3x^3 - 2x^3 = x^4 + (3x^3 - 2x^3) \][/tex]
[tex]\[ = x^4 + x^3 \][/tex]
Thus, the expression simplifies to:
[tex]\[ (x^4 + x^3)(-5x^2 + x) \][/tex]
Now, we apply the distributive property to distribute each term in [tex]\((x^4 + x^3)\)[/tex] across each term in [tex]\((-5x^2 + x)\)[/tex]:
[tex]\[ (x^4)(-5x^2) + (x^4)(x) + (x^3)(-5x^2) + (x^3)(x) \][/tex]
This means breaking it down into:
[tex]\[ (x^4 \cdot -5x^2) + (x^4 \cdot x) \][/tex]
[tex]\[ + (x^3 \cdot -5x^2) + (x^3 \cdot x) \][/tex]
Simplifying each term gives:
[tex]\[ -5x^6 + x^5 - 5x^5 + x^4 \][/tex]
Rewriting the question:
[tex]\[ \left(x^4 + x^3\right)\left(-5x^2 + x\right) = \left(x^4 + x^3\right)(-5x^2) + \left(x^4 + x^3\right)(x) \][/tex]
This shows that we've applied the distributive property, where each term in the first polynomial is individually multiplied by each term in the second polynomial.
Thus, the expression is an example of:
[tex]\[ \boxed{\text{C. The distributive property}} \][/tex]
