Extending solutions of a differential equation

Question

I have a following exercise:
Prove that every solution of an equation $x' = \sin(x^2+t^2) + 3|x|$ can be extended to all $t$ in real numbers.
How do I go about proving this? I know theorems stating that such extension is possible if the right-hand side is bounded, but here it clearly isn't.
I'll be grateful for any help. :)