Question

divide.
\\((-10x^3 + 16x^2 + 5x^4 - 12 + 13x) \div (-5x^2 + 4)\\)
write your answer in the following form: \\(\text{quotient} + \frac{\text{remainder}}{-5x^2 + 4}\\).
\\(\frac{-10x^3 + 16x^2 + 5x^4 - 12 + 13x}{-5x^2 + 4} = \square + \frac{\square}{-5x^2 + 4}\\)

Explanation:

Step1: Rearrange the dividend

First, we rearrange the terms of the dividend in descending order of powers of \(x\):
\(5x^{4}-10x^{3}+16x^{2}+13x - 12\)

Step2: Divide the leading terms

Divide the leading term of the dividend \(5x^{4}\) by the leading term of the divisor \(-5x^{2}\):
\(\frac{5x^{4}}{-5x^{2}}=-x^{2}\)

Step3: Multiply the divisor by \(-x^{2}\)

Multiply the divisor \(-5x^{2}+4\) by \(-x^{2}\):
\((-x^{2})(-5x^{2}+4)=5x^{4}-4x^{2}\)

Step4: Subtract this from the dividend

Subtract \(5x^{4}-4x^{2}\) from \(5x^{4}-10x^{3}+16x^{2}+13x - 12\):
\[

$$\begin{align*} &(5x^{4}-10x^{3}+16x^{2}+13x - 12)-(5x^{4}-4x^{2})\\ =&5x^{4}-10x^{3}+16x^{2}+13x - 12 - 5x^{4}+4x^{2}\\ =&-10x^{3}+20x^{2}+13x - 12 \end{align*}$$

\]

Step5: Divide the new leading term

Divide the leading term of the new dividend \(-10x^{3}\) by the leading term of the divisor \(-5x^{2}\):
\(\frac{-10x^{3}}{-5x^{2}} = 2x\)

Step6: Multiply the divisor by \(2x\)

Multiply the divisor \(-5x^{2}+4\) by \(2x\):
\(2x(-5x^{2}+4)=-10x^{3}+8x\)

Step7: Subtract this from the new dividend

Subtract \(-10x^{3}+8x\) from \(-10x^{3}+20x^{2}+13x - 12\):
\[

$$\begin{align*} &(-10x^{3}+20x^{2}+13x - 12)-(-10x^{3}+8x)\\ =&-10x^{3}+20x^{2}+13x - 12 + 10x^{3}-8x\\ =&20x^{2}+5x - 12 \end{align*}$$

\]

Step8: Divide the new leading term

Divide the leading term of the new dividend \(20x^{2}\) by the leading term of the divisor \(-5x^{2}\):
\(\frac{20x^{2}}{-5x^{2}}=-4\)

Step9: Multiply the divisor by \(-4\)

Multiply the divisor \(-5x^{2}+4\) by \(-4\):
\(-4(-5x^{2}+4)=20x^{2}-16\)

Step10: Subtract this from the new dividend

Subtract \(20x^{2}-16\) from \(20x^{2}+5x - 12\):
\[

$$\begin{align*} &(20x^{2}+5x - 12)-(20x^{2}-16)\\ =&20x^{2}+5x - 12 - 20x^{2}+16\\ =&5x + 4 \end{align*}$$

\]

Now, the quotient is \(-x^{2}+2x - 4\) and the remainder is \(5x + 4\).

Answer:

\(-x^{2}+2x - 4\) and the remainder is \(5x + 4\), so the expression is \(-x^{2}+2x - 4+\frac{5x + 4}{-5x^{2}+4}\)