ANSWER:
[tex]x^{2}+x-6[/tex]STEP-BY-STEP EXPLANATION:
We have the following polynomial:
[tex]4x^3+x^2-27x+18[/tex]We must divide it by 4x - 3, using the long division method, therefore:
[tex]4x^3+x^2-27x+18\div \left(4x-3\right)[/tex]We solve it below:
[tex]\begin{gathered} \text{ We divide the leading term of the dividend by the leading term of the divisor:} \\ \\ \frac{4x^3}{4x}=x^2 \\ \\ \text{ We multiply it by the divisor} \\ \\ x^2\cdot(4x-3)=4x^3-3x^2 \\ \\ \text{ We subtract the dividend from the obtained result: } \\ \\ 4x^3+x^2-27x+18-4x^3+3x^2=4x^2-27x+18 \\ \\ \text{ Finally it would be:} \\ \\ x^2+\frac{4x^2-27x+18}{4x-3} \end{gathered}[/tex]Now, we repeat the same procedure:
[tex]\begin{gathered} \text{ We divide the leading term of the dividend by the leading term of the divisor:} \\ \\ \frac{4x^2}{4x}=x \\ \\ \text{ We multiply it by the divisor} \\ \\ x\cdot(4x-3)=4x^2-3x \\ \\ \text{ We subtract the dividend from the obtained result: } \\ \\ 4x^2-27x+18-4x^2-3x=-24x+18 \\ \\ \text{ Finally it would be:} \\ \\ x^2+x+\frac{-24+18}{4x-3} \end{gathered}[/tex]We do the division for the last time and we would have the following:
[tex]\begin{gathered} \text{ We divide the leading term of the dividend by the leading term of the divisor:} \\ \\ \frac{-24x}{4x}=-6 \\ \\ \text{ We multiply it by the divisor} \\ \\ -6\cdot(4x-3)=-24x+18 \\ \\ \text{ We subtract the dividend from the obtained result: } \\ \\ -24x+18-24x+18=0 \\ \\ \text{ Finally it would be:} \\ \\ x^2+x-6 \end{gathered}[/tex]So, the correct answer is:
[tex]4x^3+x^2-27x+18\div\left(4x-3\right)=x^2+x-6[/tex]