Question

select the correct answer.
what is the quotient?
\boxed{\dfrac{4x^3 + 12x - 8}{4x}}
\bigcirc a. \\ 4x^2 + 8x - 8
\bigcirc b. \\ x + 3
\bigcirc c. \\ x^2 + 3x - 2
\bigcirc d. \\ \dfrac{x^2 + 3x - 2}{x}

Explanation:

Step1: Split the fraction

We can split the fraction \(\frac{4x^3 + 12x - 8}{4x}\) into three separate fractions: \(\frac{4x^3}{4x}+\frac{12x}{4x}-\frac{8}{4x}\).

Step2: Simplify each fraction

• For \(\frac{4x^3}{4x}\), the 4s cancel out, and using the rule of exponents \(x^a\div x^b = x^{a - b}\), we get \(x^{3-1}=x^2\).
• For \(\frac{12x}{4x}\), the \(x\)s cancel out, and \(\frac{12}{4}=3\), so this simplifies to 3.
• For \(\frac{8}{4x}\), we can simplify \(\frac{8}{4}\) to 2, so it becomes \(\frac{2}{x}\)? Wait, no, wait, the original numerator is \(4x^3 + 12x - 8\), denominator is \(4x\). Wait, I made a mistake. Wait, the numerator is \(4x^3+12x - 8\)? Wait, no, looking at the options, maybe the original problem is \(\frac{4x^3 + 12x^2 - 8x}{4x}\)? Wait, the user's problem has a typo? Wait, the options have \(x^2 + 3x - 2\) as option C. Let me re - examine.

Wait, maybe the original numerator is \(4x^3+12x^2 - 8x\). Let's assume that (maybe a typo in the problem statement, because with \(4x^3 + 12x - 8\) and denominator \(4x\), we can't get the options. So let's correct the numerator to \(4x^3+12x^2 - 8x\)).

So splitting \(\frac{4x^3 + 12x^2 - 8x}{4x}=\frac{4x^3}{4x}+\frac{12x^2}{4x}-\frac{8x}{4x}\)

• \(\frac{4x^3}{4x}=x^2\) (4s cancel, \(x^3\div x = x^{2}\))
• \(\frac{12x^2}{4x}=3x\) (12/4 = 3, \(x^2\div x=x\))
• \(\frac{8x}{4x}=2\) (8/4 = 2, \(x\div x = 1\))

So combining these, we get \(x^2+3x - 2\).

Wait, maybe the original problem's numerator was \(4x^3+12x^2 - 8x\) and denominator \(4x\). Then the quotient is \(x^2 + 3x - 2\), which is option C.

Answer:

C. \(x^{2}+3x - 2\)