The transformed equation of $x^{4} + 8 x^{3} + x - 5 = 0$ by eliminating second term is

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$x^{4} + 24 x^{2} + 65 x + 55 = 0$

$x^{4} - 24 x^{2} - 65 x + 55 = 0$

$x^{4} + 24 x^{2} + 65 x - 55 = 0$

$x^{4} - 24 x^{2} + 65 x - 55 = 0$

answer is A.

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Detailed Solution

$\begin{array}{l} f (x) = x^{4} + 8 x^{3} + x - 5 = 0 \\ L e t n = 4, r = 2 \\ ∴ f^{(n - r + 1)} (h) = 0 \Rightarrow f^{3} (h) = 0 \\ f^{1} (x) = 4 x^{3} + 24 x^{2} + 1 \\ f^{''} (x) = 12 x^{2} + 48 x \\ f^{'''} (x) = 24 x + 48 \\ f^{'''} (h) = 0 \Rightarrow 24 h + 48 = 0 \Rightarrow h = - 2 \\ Re q u i r e d E q u a t i o n i s f (x - 2) = 0 \\ \Rightarrow {(x - 2)}^{4} + 8 {(x - 2)}^{3} + (x - 2) - 5 = 0 \\ \Rightarrow {(x^{2} + 4 - 4 x)}^{2} + 8 (x^{3} - 6 x^{2} + 12 x - 8) + x - 7 = 0 \\ \Rightarrow x^{4} + 16 + 16 x^{2} + 8 x^{2} - 32 x - 8 x^{3} + 8 x^{3} - 48 x^{2} + 96 x - 64 + x - 7 = 0 \\ \Rightarrow x^{4} - 24 x^{2} + 65 x - 55 = 0 \end{array}$