A carpenter was hired to build 192 window frames.

We have:

$S=192, a=5, d=2$

Now,

$S_{n}=192$

$\Rightarrow \frac{n}{2}[2 a+(n-1) d]=192$

$\Rightarrow \frac{n}{2}[2 \times 5+(n-1) \times 2]=192$

$\Rightarrow \frac{n}{2}[10+2 n-2]=192$

$\Rightarrow \frac{n}{2}[2 n+8]=192$

$\Rightarrow n(n+4)=192$

$\Rightarrow n^{2}+4 n=192$

$\Rightarrow n^{2}-12 n+16 n-192=0$

$\Rightarrow n(n-12)+16(n-12)=0$

$\Rightarrow(n-12)(n+16)=0$

$\Rightarrow(n-12)=0$ or $(n+16)=0$

$\Rightarrow n=12$ or $n=-16$

$\because n$ cannot be negative.

$\therefore n=12$

So, the carpenter takes 12 days to finish the job.