If a and b are the roots of the equation x ^ 2 - 7x - 1 = 0 , then...

Since, $a$ and $b$ are roots of ${x}^{2} - 7x - 1 = 0$, so

${a}^{2} - 7a - 1 = 0$

$⇒ {a}^{2} - 1 = 7a$

$⇒ {a}^{4} + 1 - 2{a}^{2} = 49{a}^{2}$

$⇒ {a}^{4} + 1 = 51{a}^{2} . . . \left(1\right)$

Similarly,

${b}^{4} + 1 = 51{b}^{2} . . . \left(2\right)$

Now,

$\dfrac{{a}^{21} + {b}^{21} + {a}^{17} + {b}^{17}}{{a}^{19} + {b}^{19}}$

$= \dfrac{{a}^{17}\left({a}^{4} + 1\right) + {b}^{17}\left({b}^{4} + 1\right)}{{a}^{19} + {b}^{19}}$

$= \dfrac{51{a}^{19} + 51{b}^{19}}{{a}^{19} + {b}^{19}}$

$= 51$