Let's analyze each option to determine which demonstrates the correct process of polynomial division by recognizing it as the inverse operation of multiplication.
1. Option 1:
[tex]\[
\frac{N x^2+1 x+12}{4 x}=\left(-\frac{1}{4 x}\right)\left(8 x^2-4 x+12\right)
\][/tex]
This option suggests that dividing by [tex]\(4x\)[/tex] is the same as multiplying by [tex]\(-\frac{1}{4x}\)[/tex]. However, the correct inverse operation involves multiplying by [tex]\(\frac{1}{4x}\)[/tex], not [tex]\(-\frac{1}{4x}\)[/tex]. Therefore, this option is incorrect.
2. Option 2:
[tex]\[
\frac{8 x^2-4 x+12}{4 x}-(-4 x)\left(8 x^2-4 x+12\right)
\][/tex]
This option is not in the form that demonstrates division by recognizing it as a multiplication operation. It incorrectly mixes subtraction and potential multiplication, which is not what we are looking for. Therefore, this option is also incorrect.
3. Option 3:
[tex]\[
\frac{k x^2-4 x+12}{4 x}=\left(\frac{1}{4 x}\right)\left(8 x^2-4 x+12\right)
\][/tex]
Here, dividing [tex]\(k x^2 - 4 x + 12\)[/tex] by [tex]\(4x\)[/tex] is shown to be the same as multiplying [tex]\((8 x^2 - 4 x + 12)\)[/tex] by [tex]\(\frac{1}{4x}\)[/tex]. This form correctly demonstrates the inverse relationship. Hence, this option is the correct one.
4. Option 4:
[tex]\[
\frac{8 x^2-4 x+12}{4 x}=(4 x)\left(8 x^2-4 x+12\right)
\][/tex]
This option suggests dividing a polynomial by [tex]\(4x\)[/tex] is equivalent to multiplying the polynomial by [tex]\(4x\)[/tex], which is incorrect. Therefore, this option is incorrect too.
Given our analysis, we conclude that Option 3 correctly demonstrates polynomial division as the inverse of multiplication, leading us to the correct option:
[tex]\[
\frac{k x^2-4 x+12}{4 x}=\left(\frac{1}{4 x}\right)\left(8 x^2-4 x+12\right)
\][/tex]
Therefore, the correct choice is indeed Option 3.
