Use the table to answer the question.

\begin{tabular}{|c|c|}
\hline
[tex]$-4x^4$[/tex] & [tex]$10x$[/tex] \\
\hline
[tex]$-20x^7$[/tex] & [tex]$50x^4$[/tex] \\
\hline
[tex]$4x^4$[/tex] & [tex]$-10x$[/tex] \\
\hline
\end{tabular}

Find the product when you multiply the polynomials [tex]$\left(-4x^4 + 10x\right)\left(5x^3 - 1\right)$[/tex].

A. [tex]$20x^7 + 54x^4 - 10x$[/tex]

B. [tex]$-20x^7 - 54x^4 - 10x$[/tex]

C. [tex]$-20x^7 + 54x^4 - 10x$[/tex]

D. [tex]$-20x^7 + 50x^4 - 6x$[/tex]



Answer :

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]