<s>[INST] What does it mean to solve or find solutions in mathematics? Something that has been really confusing me lately is that this equation has four solutions $$3x(x+1)(x^2+x+2)=16x(x+1)(2x+1)$$
But what does that mean? Until now solutions to me has meant, what are the coordinates of $x$ when $y$ equals a given value, normally $y=0$.
But this equation has kind of thrown me off because when plotting this equation on a graph I get only two points at which the line crosses the $x$-axis.
I hope someone understands my point.
Regards
 [/INST] Definition: Solve. We say an equation is solved if and only if we have listed the set of all objects which make the equation true. We say  we have found a solution if  we have found an element of the solution set. 
Example: If $x= 0$, then 
\begin{align}
3x(x+1)(x^2+x+2)=16x(x+1)(2x+1)
\end{align}
becomes
\begin{align}
0 = 0
\end{align}
which is true, and hence $x =0$ solves the equation.
Example: If $x =1$, then
\begin{align}
3x(x+1)(x^2+x+2)=16x(x+1)(2x+1)
\end{align}
becomes
\begin{align}
3\cdot 2 \cdot 4 = 16\cdot 2\cdot 3
\end{align}
which is false, and hence $x=1$ is not a solution.
</s>