We have to find the continued fraction expansion of the roots of $1553 t^2 + 6014 t + 5820 = 0$, that is, $(\sqrt{14356} - 6014) / 3106$
Simplifying, $(\sqrt{3589}- 3007) / 1553$
The continued fraction expansion is: -2+ //9, 1, 3, 2, 2, 9, 1, 1, 2, 1, 4, 13, 9, 1, 9, 1, 118, 1, 9, 1, 9, 13, 4, 1, 2, 1, 1, 9, 2, 2, 4, 29, 1, 2, 1, 1, 1, 39, 3, 3, 3, 3, 39, 1, 1, 1, 2, 1, 29, 4// where the periodic part is marked in bold (the period has 47 coefficients).
Where is this expansion coming from? I put the sqrt expression into Wolfram Alpha and it gives me a different expansion.
The difference is that Alpha is giving you a negative continued fraction-note that it is $- [1; 1, 8, 1, 3, 2, 2, 9, \dots]$ Your answer key is giving you a positive continued fraction that is added to $-2$. If you add $3106$ to the numerator to make the value positive, Alpha gives $[0; 9, 1, 3, \overline{2, 2, 9, 1, 1, 2, 1, 4, 13, 9, 1, 9, 1, 118, 1, 9, 1, 9, 13, 4, 1, 2, 1, 1, 9, 2, 2, 4, 29, 1, 2, 1, 1, 1, 39, 3, 3, 3, 3, 39, 1, 1, 1, 2, 1, 29, 4}]$ as well