Sure, let's solve the problem by multiplying the polynomials [tex]\((-4x^4 + 10x)\)[/tex] and [tex]\((5x^3 - 1)\)[/tex] step-by-step.
First, we'll use the distributive property to multiply each term in the first polynomial by each term in the second polynomial.
The terms in [tex]\((-4x^4 + 10x)\)[/tex] are:
1. [tex]\(-4x^4\)[/tex]
2. [tex]\(10x\)[/tex]
The terms in [tex]\((5x^3 - 1)\)[/tex] are:
1. [tex]\(5x^3\)[/tex]
2. [tex]\(-1\)[/tex]
Now, perform the multiplication for each combination of terms:
1. [tex]\((-4x^4) \cdot (5x^3)\)[/tex]:
[tex]\[
-4x^4 \cdot 5x^3 = -20x^7
\][/tex]
2. [tex]\((-4x^4) \cdot (-1)\)[/tex]:
[tex]\[
-4x^4 \cdot -1 = 4x^4
\][/tex]
3. [tex]\((10x) \cdot (5x^3)\)[/tex]:
[tex]\[
10x \cdot 5x^3 = 50x^4
\][/tex]
4. [tex]\((10x) \cdot (-1)\)[/tex]:
[tex]\[
10x \cdot -1 = -10x
\][/tex]
Let's combine these results to write the polynomial:
[tex]\[
-20x^7 + 4x^4 + 50x^4 - 10x
\][/tex]
Next, combine the like terms. The like terms in this polynomial are [tex]\(4x^4\)[/tex] and [tex]\(50x^4\)[/tex]:
[tex]\[
-20x^7 + (4x^4 + 50x^4) - 10x
\][/tex]
Combine [tex]\(4x^4\)[/tex] and [tex]\(50x^4\)[/tex]:
[tex]\[
4x^4 + 50x^4 = 54x^4
\][/tex]
So, the final polynomial expression is:
[tex]\[
-20x^7 + 54x^4 - 10x
\][/tex]
Therefore, the correct answer is:
[tex]\[
-20x^7 + 54x^4 - 10x
\][/tex]
So, the product of the polynomials [tex]\((-4x^4 + 10x)\)[/tex] and [tex]\((5x^3 - 1)\)[/tex] is:
[tex]\[
\boxed{-20x^7 + 54x^4 - 10x}
\][/tex]
Thus, the correct option from the given choices is:
[tex]\(
-20x^7 + 54x^4 - 10x
\)[/tex]
