Q. Find the quotient. \( (5x^{4} – 3x^{2} + 4) \div (x + 1) \)

Problem: Divide: \( \dfrac{5x^{4}-3x^{2}+4}{x+1} \)

Step 1 – Set up the dividend with placeholder terms

Write the dividend with zero coefficients for missing powers:
\[
5x^{4}+0x^{3}-3x^{2}+0x+4
\]
The divisor is \(x+1\).

Step 2 – Divide the leading terms

\[
\frac{5x^{4}}{x}=5x^{3}
\]
This is the first term of the quotient.

Step 3 – Multiply and subtract

Multiply \(5x^{3}\) by \(x+1\):
\[
5x^{3}\cdot(x+1)=5x^{4}+5x^{3}
\]
Subtract from the dividend (first two terms):
\[
(5x^{4}+0x^{3})-(5x^{4}+5x^{3})=-5x^{3}
\]
Bring down the next term \(-3x^{2}\) to form the new partial remainder:
\[
-5x^{3}-3x^{2}
\]

Step 4 – Divide the new leading term

\[
\frac{-5x^{3}}{x}=-5x^{2}
\]
This is the second term of the quotient.

Step 5 – Multiply and subtract again

Multiply \(-5x^{2}\) by \(x+1\):
\[
-5x^{2}\cdot(x+1)=-5x^{3}-5x^{2}
\]
Subtract from the partial remainder:
\[
(-5x^{3}-3x^{2})-(-5x^{3}-5x^{2})=2x^{2}
\]
Bring down the next term \(0x\) to get \(2x^{2}+0x\).

Step 6 – Divide the next leading term

\[
\frac{2x^{2}}{x}=2x
\]
This is the third term of the quotient.

Step 7 – Multiply and subtract

Multiply \(2x\) by \(x+1\):
\[
2x\cdot(x+1)=2x^{2}+2x
\]
Subtract from the partial remainder:
\[
(2x^{2}+0x)-(2x^{2}+2x)=-2x
\]
Bring down the constant \(+4\) to get \(-2x+4\).

Step 8 – Divide the final leading term

\[
\frac{-2x}{x}=-2
\]
This is the final term of the quotient.

Step 9 – Final multiply and remainder

Multiply \(-2\) by \(x+1\):
\[
-2\cdot(x+1)=-2x-2
\]
Subtract:
\[
(-2x+4)-(-2x-2)=6
\]
The remainder is \(6\).

Answer

Combine quotient and remainder over the divisor:
\[
\frac{5x^{4}-3x^{2}+4}{x+1}=5x^{3}-5x^{2}+2x-2+\frac{6}{x+1}
\]