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(x^(4)-x^(2)-5)÷(x^(2)+4x-1) A) x^(2)-4x+4 B) x^(2)+4x-4 C) x^(2)-4x+16+(-68 x+11)/(x^(2)+4x-1) D) x^(2)+4x-4+(-5x-1)/(x^(2)+4x-1) E) x^(2)-4x+4-(4)/(x^(2)-4x+4)

(x4−x2−5)÷(x2+4x−1) \left(x^{4}-x^{2}-5\right) \div\left(x^{2}+4 x-1\right) \newlineA) x2−4x+4 x^{2}-4 x+4 \newlineB) x2+4x−4 x^{2}+4 x-4 \newlineC) x2−4x+16+−68x+11x2+4x−1 x^{2}-4 x+16+\frac{-68 x+11}{x^{2}+4 x-1} \newlineD) x2+4x−4+−5x−1x2+4x−1 x^{2}+4 x-4+\frac{-5 x-1}{x^{2}+4 x-1} \newlineE) x2−4x+4−4x2−4x+4 x^{2}-4 x+4-\frac{4}{x^{2}-4 x+4}

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Q. (x4−x2−5)÷(x2+4x−1) \left(x^{4}-x^{2}-5\right) \div\left(x^{2}+4 x-1\right) \newlineA) x2−4x+4 x^{2}-4 x+4 \newlineB) x2+4x−4 x^{2}+4 x-4 \newlineC) x2−4x+16+−68x+11x2+4x−1 x^{2}-4 x+16+\frac{-68 x+11}{x^{2}+4 x-1} \newlineD) x2+4x−4+−5x−1x2+4x−1 x^{2}+4 x-4+\frac{-5 x-1}{x^{2}+4 x-1} \newlineE) x2−4x+4−4x2−4x+4 x^{2}-4 x+4-\frac{4}{x^{2}-4 x+4}
  1. Divide Leading Terms: To solve this problem, we will perform polynomial long division. We will divide the polynomial in the numerator, x4−x2−5x^{4}-x^{2}-5, by the polynomial in the denominator, x2+4x−1x^{2}+4x-1.
  2. Subtract and Simplify: First, we divide the leading term of the numerator, x4x^{4}, by the leading term of the denominator, x2x^{2}, to get x2x^{2}. This will be the first term of our quotient.
  3. Divide New Leading Term: Next, we multiply the entire denominator, (x2+4x−1)(x^{2}+4x-1), by the term we just found, x2x^{2}, and subtract this from the numerator.\newline(x4−x2−5)−(x2⋅(x2+4x−1))=(x4−x2−5)−(x4+4x3−x2)(x^{4}-x^{2}-5) - (x^{2} \cdot (x^{2}+4x-1)) = (x^{4}-x^{2}-5) - (x^{4}+4x^3-x^2)
  4. Subtract and Simplify: Simplify the subtraction by combining like terms. \newline(x4−x2−5)−(x4+4x3−x2)=−4x3−5(x^{4}-x^{2}-5) - (x^{4}+4x^3-x^2) = -4x^3-5
  5. Divide New Leading Term: Now, we divide the new leading term of the remainder, −4x3-4x^3, by the leading term of the denominator, x2x^{2}, to get −4x-4x. This will be the next term of our quotient.
  6. Subtract and Simplify: We multiply the entire denominator, (x2+4x−1)(x^{2}+4x-1), by the term we just found, −4x-4x, and subtract this from the current remainder.\newline−4x3−5−(−4x×(x2+4x−1))=−4x3−5−(−4x3−16x2+4x)-4x^3-5 - (-4x \times (x^{2}+4x-1)) = -4x^3-5 - (-4x^3-16x^2+4x)
  7. Final Remainder: Simplify the subtraction by combining like terms. \newline−4x3−5−(−4x3−16x2+4x)=16x2−4x−5-4x^3-5 - (-4x^3-16x^2+4x) = 16x^2-4x-5
  8. Final Remainder: Simplify the subtraction by combining like terms. \newline−4x3−5−(−4x3−16x2+4x)=16x2−4x−5-4x^3-5 - (-4x^3-16x^2+4x) = 16x^2-4x-5Now, we divide the new leading term of the remainder, 16x216x^2, by the leading term of the denominator, x2x^{2}, to get 1616. This will be the next term of our quotient.
  9. Final Remainder: Simplify the subtraction by combining like terms. \newline−4x3−5−(−4x3−16x2+4x)=16x2−4x−5-4x^3-5 - (-4x^3-16x^2+4x) = 16x^2-4x-5Now, we divide the new leading term of the remainder, 16x216x^2, by the leading term of the denominator, x2x^{2}, to get 1616. This will be the next term of our quotient.We multiply the entire denominator, (x2+4x−1)(x^{2}+4x-1), by the term we just found, 1616, and subtract this from the current remainder.\newline16x2−4x−5−(16∗(x2+4x−1))=16x2−4x−5−(16x2+64x−16)16x^2-4x-5 - (16 * (x^{2}+4x-1)) = 16x^2-4x-5 - (16x^2+64x-16)
  10. Final Remainder: Simplify the subtraction by combining like terms. \newline−4x3−5−(−4x3−16x2+4x)=16x2−4x−5-4x^3-5 - (-4x^3-16x^2+4x) = 16x^2-4x-5Now, we divide the new leading term of the remainder, 16x216x^2, by the leading term of the denominator, x2x^{2}, to get 1616. This will be the next term of our quotient.We multiply the entire denominator, (x2+4x−1)(x^{2}+4x-1), by the term we just found, 1616, and subtract this from the current remainder.\newline16x2−4x−5−(16×(x2+4x−1))=16x2−4x−5−(16x2+64x−16)16x^2-4x-5 - (16 \times (x^{2}+4x-1)) = 16x^2-4x-5 - (16x^2+64x-16)Simplify the subtraction by combining like terms.\newline16x2−4x−5−(16x2+64x−16)=−68x+1116x^2-4x-5 - (16x^2+64x-16) = -68x+11
  11. Final Remainder: Simplify the subtraction by combining like terms. \newline−4x3−5−(−4x3−16x2+4x)=16x2−4x−5-4x^3-5 - (-4x^3-16x^2+4x) = 16x^2-4x-5Now, we divide the new leading term of the remainder, 16x216x^2, by the leading term of the denominator, x2x^{2}, to get 1616. This will be the next term of our quotient.We multiply the entire denominator, (x2+4x−1)(x^{2}+4x-1), by the term we just found, 1616, and subtract this from the current remainder.\newline16x2−4x−5−(16×(x2+4x−1))=16x2−4x−5−(16x2+64x−16)16x^2-4x-5 - (16 \times (x^{2}+4x-1)) = 16x^2-4x-5 - (16x^2+64x-16)Simplify the subtraction by combining like terms.\newline16x2−4x−5−(16x2+64x−16)=−68x+1116x^2-4x-5 - (16x^2+64x-16) = -68x+11The remainder is now −68x+11-68x+11, which cannot be divided by the denominator x2+4x−1x^{2}+4x-1 since its degree is lower than the degree of the denominator. Therefore, the remainder will be written as a fraction over the original denominator.
  12. Final Remainder: Simplify the subtraction by combining like terms.\newline−4x3−5−(−4x3−16x2+4x)=16x2−4x−5-4x^3-5 - (-4x^3-16x^2+4x) = 16x^2-4x-5Now, we divide the new leading term of the remainder, 16x216x^2, by the leading term of the denominator, x2x^{2}, to get 1616. This will be the next term of our quotient.We multiply the entire denominator, (x2+4x−1)(x^{2}+4x-1), by the term we just found, 1616, and subtract this from the current remainder.\newline16x2−4x−5−(16×(x2+4x−1))=16x2−4x−5−(16x2+64x−16)16x^2-4x-5 - (16 \times (x^{2}+4x-1)) = 16x^2-4x-5 - (16x^2+64x-16)Simplify the subtraction by combining like terms.\newline16x2−4x−5−(16x2+64x−16)=−68x+1116x^2-4x-5 - (16x^2+64x-16) = -68x+11The remainder is now −68x+11-68x+11, which cannot be divided by the denominator x2+4x−1x^{2}+4x-1 since its degree is lower than the degree of the denominator. Therefore, the remainder will be written as a fraction over the original denominator.Combine all the terms we found for the quotient and write the remainder as a fraction over the denominator to get the final answer.\newline16x216x^200

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