Question 7
Let $a,$ $b,$ $c,$ and $d$ be real numbers that satisfy the system of equations $$
\begin{aligned} a + b &= -3, \\ ab + bc + ca &= -4, \end{aligned}
$$ $$
\begin{aligned} abc + bcd + cda + dab &= 14, \\ abcd &= 30. \end{aligned}
$$ There exist relatively prime positive integers $m$ and $n$ such that $$
a^2 + b^2 + c^2 + d^2 = \frac{m}{n}.
$$ Find $m + n.$
\begin{aligned} a + b &= -3, \\ ab + bc + ca &= -4, \end{aligned}
$$ $$
\begin{aligned} abc + bcd + cda + dab &= 14, \\ abcd &= 30. \end{aligned}
$$ There exist relatively prime positive integers $m$ and $n$ such that $$
a^2 + b^2 + c^2 + d^2 = \frac{m}{n}.
$$ Find $m + n.$