Answer: 1649/36
Explanation:
If the roots are r and s, the sum and product of the roots are known: r+s = -\frac{21}{18} = -\frac76 and rs = \frac{-400}{18} = -\frac{200}{9}. The goal is to calculate $r^2 + s^2$, but this can be done using the known information.
\[r^2 +s^2 = (r+s)^2 - 2rs = \left(-\frac76\right)^2 - 2\left(-\frac{200}{9}\right)
= \frac{49}{36} + \frac{400}{9} = \boxed{\frac{1649}{36}}.\]