Divide the leading term of the dividend (x^4) by the leading term of the divisor (x²) to get x². Multiply the divisor by x² and subtract the result from the dividend. Bring down the remaining terms.

Divide the leading term of the dividend (x^4) by the leading term of the divisor (x²) to get x². Multiply the divisor by x² and subtract the result from the dividend. Bring down the remaining terms.

Now, divide the leading term of the new polynomial (2x³) by the leading term of the divisor (x²) to get 2x. Multiply the divisor by 2x and subtract the result. Bring down the next term.

Divide x⁴ + 2x³ + 2x² + $2x + 2$ by (x² − 1), and write down the quotient and the remainder.

Answer: Quotient: $x^2 + 2x + 3$ , Remainder: $4x + 5$

Step-by-step solution

Step 1: Set up for polynomial long division

We need to perform polynomial long division. First, we write the dividend and the divisor in the standard long division format. Ensure all powers of $x$ are present in the dividend, using zero coefficients if necessary.

Step 2: Perform the first step of division

Divide the leading term of the dividend ( $x^4$ ) by the leading term of the divisor ( $x^2$ ) to get $x^2$ . Multiply the divisor by $x^2$ and subtract the result from the dividend. Bring down the remaining terms.

Step 3: Perform the second step of division

Now, divide the leading term of the new polynomial ( $2x^3$ ) by the leading term of the divisor ( $x^2$ ) to get $2x$ . Multiply the divisor by $2x$ and subtract the result. Bring down the next term.

Step 4: Perform the third step of division

Finally, divide the leading term of the current polynomial ( $3x^2$ ) by the leading term of the divisor ( $x^2$ ) to get $3$ . Multiply the divisor by $3$ and subtract the result. The degree of the remaining polynomial ( $4x+5$ ) is less than the degree of the divisor ( $x^2-1$ ), so this is our remainder.

Step 5: State the quotient and remainder

From the long division process, we have determined the quotient and the remainder.

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Divide x⁴ + 2x³ + 2x² + 2x+22x + 22x+2 by (x² − 1), and write down the quotient and the remainder.