How do you solve the following nonlinear equations? How to solve the following system of equations

$$
\begin{cases}
a+c=12\\
b+ac+d=86\\
bc+ad=300\\
bd=625\\
\end{cases}
$$


Well, $b=\cfrac {625}{d}$ from your last equation, and $a=-c+12$ from your first equation.
Substitute $b$ and $a$ into your second equation to get $\cfrac{625}{d}+(-c+12)(c)+d=86$
Do the same for your third equation.
