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Answer 1

$6) \: A \: Polynomial \: of \: the \: form \:  p(x) \: = \: ax ^ {2}  + bx + c, \\ \: where \:  a  \: 0 \: and  \: a, \:  b,  \: c are \: real \: numbers \: \\ and \:  x  \: is \: a \: real \: variable \: is \: called \: a  \: quadratic \: polynomial. \\ \: so \: the \: answer \: is \: option \: b$

$7) \: let(p) = {4x}^{3} + {3x}^{2} - 4x + k \\ given \: (x - 1) \: is \: a \: factor \: of \: p(x) \\ x - 1 = 0 \\ x = 1 \\ \therefore \: 4 \times {(1)}^{2} + 3 \times {(1)}^{2} - 4 \times 1 + k = 0 \\ 4 + 3 - 4 + k = 0 \\ k = - 3 \\ so \: the \: correct \: option \: is \: c \: \: \: i.e. \: - 3$

$8) \: x + \sqrt{ {x}^{2} + 1 } \\ \sqrt{ {x}^{2} + 1 } = - x \\ { (\sqrt{ {x}^{2} + 1} })^{2} = {( - x)}^{2} \\ {x}^{2} + 1 = {x}^{2} \\ by \: cancelling \: {x}^{2} \: in \: both \: sides \: we \: get \: 1 \\ so \: the \: correct \: answer \: is \: option \: c \:$

$9) \: {(3)}^{3} + {(5)}^{3} - {( 8)}^{3} = 0 \\ 27 + 125 - 512 = 0$

$152 - 512 = 0$

$- 360$

$so \: the \: correct \: option \: is \: no. \: b$

Answer 2

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$Answer$

Refer to the above attachment ⬆️⬆️.