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Answer :
We begin with the expression
[tex]$$
(x - 4)\left(-4x^3 + 9x + 3\right).
$$[/tex]
Step 1. Distribute the terms
Distribute by multiplying [tex]$(x - 4)$[/tex] with each term in the second factor:
[tex]\[
\begin{aligned}
(x - 4)(-4x^3 + 9x + 3) &= x(-4x^3 + 9x + 3) - 4(-4x^3 + 9x + 3) \\
&= \text{(Term 1)} + \text{(Term 2)}.
\end{aligned}
\][/tex]
Step 2. Multiply each part
1. Multiply [tex]$x$[/tex] by each term inside the parentheses:
[tex]\[
x(-4x^3) = -4x^4,\quad x(9x) = 9x^2,\quad x(3) = 3x.
\][/tex]
So, the product is:
[tex]$$
-4x^4 + 9x^2 + 3x.
$$[/tex]
2. Multiply [tex]$-4$[/tex] by each term inside the parentheses:
[tex]\[
-4(-4x^3) = 16x^3,\quad -4(9x) = -36x,\quad -4(3) = -12.
\][/tex]
This gives:
[tex]$$
16x^3 - 36x - 12.
$$[/tex]
Step 3. Combine like terms
Now, add the two results:
[tex]\[
\begin{aligned}
-4x^4 + 9x^2 + 3x &+ 16x^3 - 36x - 12 \\
&= -4x^4 + 16x^3 + 9x^2 + (3x - 36x) - 12.
\end{aligned}
\][/tex]
Combine the [tex]$x$[/tex]-terms:
[tex]\[
3x - 36x = -33x.
\][/tex]
Thus, the simplified expression is:
[tex]$$
-4x^4 + 16x^3 + 9x^2 - 33x - 12.
$$[/tex]
Final Answer
The simplified expression is:
[tex]$$
-4x^4 + 16x^3 + 9x^2 - 33x - 12.
$$[/tex]
[tex]$$
(x - 4)\left(-4x^3 + 9x + 3\right).
$$[/tex]
Step 1. Distribute the terms
Distribute by multiplying [tex]$(x - 4)$[/tex] with each term in the second factor:
[tex]\[
\begin{aligned}
(x - 4)(-4x^3 + 9x + 3) &= x(-4x^3 + 9x + 3) - 4(-4x^3 + 9x + 3) \\
&= \text{(Term 1)} + \text{(Term 2)}.
\end{aligned}
\][/tex]
Step 2. Multiply each part
1. Multiply [tex]$x$[/tex] by each term inside the parentheses:
[tex]\[
x(-4x^3) = -4x^4,\quad x(9x) = 9x^2,\quad x(3) = 3x.
\][/tex]
So, the product is:
[tex]$$
-4x^4 + 9x^2 + 3x.
$$[/tex]
2. Multiply [tex]$-4$[/tex] by each term inside the parentheses:
[tex]\[
-4(-4x^3) = 16x^3,\quad -4(9x) = -36x,\quad -4(3) = -12.
\][/tex]
This gives:
[tex]$$
16x^3 - 36x - 12.
$$[/tex]
Step 3. Combine like terms
Now, add the two results:
[tex]\[
\begin{aligned}
-4x^4 + 9x^2 + 3x &+ 16x^3 - 36x - 12 \\
&= -4x^4 + 16x^3 + 9x^2 + (3x - 36x) - 12.
\end{aligned}
\][/tex]
Combine the [tex]$x$[/tex]-terms:
[tex]\[
3x - 36x = -33x.
\][/tex]
Thus, the simplified expression is:
[tex]$$
-4x^4 + 16x^3 + 9x^2 - 33x - 12.
$$[/tex]
Final Answer
The simplified expression is:
[tex]$$
-4x^4 + 16x^3 + 9x^2 - 33x - 12.
$$[/tex]
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