[tex]f(x)=\left\{ \begin{array}{cc} ax^4-2x+3,& \mbox{ if } {0<x<12}\\ bx^2+3, & \mbox{ if } {12<x<15}\end{array}\right.[/tex]
Two errors indicated in red. You missed the "\" before beginning array and you need a period at the very end (for some reason you need it when dealing with \left\{ ... \right. )
This gives :
Personally, I'd align it to the left (replacing the c's with l's):