Questions: Divide. [ (-12 x^3-4 x^2-2 x^4+18+15 x) div (-2 x^2+4) ] Write your answer in the following form: Quotient +frac Remainder -2 x^2+4. [ frac-12 x^3-4 x^2-2 x^4+18+15 x-2 x^2+4=square+fracsquare-2 x^2+4 ]

$Divide. [ (-12 x^3-4 x^2-2 x^4+18+15 x) div (-2 x^2+4) ] Write your answer in the following form: Quotient +frac Remainder -2 x^2+4. [ frac-12 x^3-4 x^2-2 x^4+18+15 x-2 x^2+4=square+fracsquare-2 x^2+4 ]$

Transcript text: Divide. \[ \left(-12 x^{3}-4 x^{2}-2 x^{4}+18+15 x\right) \div\left(-2 x^{2}+4\right) \] Write your answer in the following form: Quotient $+\frac{\text { Remainder }}{-2 x^{2}+4}$. \[ \frac{-12 x^{3}-4 x^{2}-2 x^{4}+18+15 x}{-2 x^{2}+4}=\square+\frac{\square}{-2 x^{2}+4} \]

Solution

Solution Steps

Step 1: Polynomial Long Division

We start by dividing the polynomial $-2 x^{4} - 12 x^{3} - 4 x^{2} + 15 x + 18$ by $-2 x^{2} + 4$.

Divide $-2 x^{4}$ by $-2 x^{2}$, resulting in $x^{2}$, with a remainder of $-12 x^{3} - 8 x^{2} + 15 x + 18$.
Next, divide $-12 x^{3}$ by $-2 x^{2}$, resulting in $6 x$, with a remainder of $-8 x^{2} - 9 x + 18$.
Finally, divide $-8 x^{2}$ by $-2 x^{2}$, resulting in $4$, with a remainder of $2 - 9 x$.

Step 2: Constructing the Quotient and Remainder

From the division process, we find:

Quotient: $x^{2} + 6 x + 4$
Remainder: $2 - 9 x$

Step 3: Final Expression

The complete division can be expressed as: \[ \frac{-2 x^{4} - 12 x^{3} - 4 x^{2} + 15 x + 18}{-2 x^{2} + 4} = x^{2} + 6 x + 4 + \frac{2 - 9 x}{-2 x^{2} + 4} \]