High School

Simplify the expression:

[tex]\[
(x-4)\left(-4x^3+9x+3\right)
\][/tex]

Choose the correct simplified form:

A. [tex]-4x^4+16x^3+9x^2-33x-12[/tex]

B. [tex]-4x^4-9x^2-39x-12[/tex]

C. [tex]16x^4+16x^3+9x^2+7[/tex]

D. [tex]16x^4-9x^2+7[/tex]

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]

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