Although we do find the roots, the following is mainly a spoof of school algebra.
In school algebra, students are expected to solve very special equations of the form $ax^2+bx +c=0$, where $a$, $b$, and $c$ are integers, by factoring. The Quadratic Formula, and even the Rational Roots Theorem, are withheld from them, as that would make the problem too simple.
The process they are taught involves factoring $a$ and $c$, and fiddling a bit to try to produce $-b$. They are only given quadratics that yield to this process.
Let's play that game with our equation, to see whether we are dealing with a variant of a school problem. So we factor $24+7i$ in the Gaussian integers. Note that $(24+7i)(24-7i)=625$. If you have done some computations with Gaussian integers, you will see that the Gaussian primes involved in the factorization are $2\pm i$ (and associates, but we needn't worry about these). Also, since $5$ does not divide $24+7i$, we know that $24+7i$ must be an associate of $(2\pm i)^4$. Pretty quickly we find that $24+7i=-i(2+i)^4$.
Now let's find two Gaussian integers whose product is $-i(2+i)^4$ and whose sum is $7+i$. Note that $(2+i)^2=3+4i$ and $-i(2+i)^2=4-3i$. Their sum is $7+i$, so we have found the roots.