Answer
$\dfrac{3 \sin{\theta}+1}{\sin{\theta}+1}$
Work Step by Step
Factor each trinomial:
$3 \sin^2{\theta}+4 \sin{\theta}+1 = (\sin{\theta}+1)(3 \sin{\theta}+1)$
$\sin^2{\theta}+2 \sin{\theta}+1 = (\sin{\theta}+1)(\sin{\theta}+1)$
Thus,
$\dfrac{3 \sin^2{\theta}+4 \sin{\theta}+1}{\sin^2{\theta}+2 \sin{\theta}+1} = \dfrac{(\sin{\theta}+1)(3 \sin{\theta}+1)}{(\sin{\theta}+1)(\sin{\theta}+1)}$
Cancel the common factorss to obtain:
$\dfrac{(\sin{\theta}+1)(3 \sin{\theta}+1)}{(\sin{\theta}+1)(\sin{\theta}+1)} = \boxed{\dfrac{3 \sin{\theta}+1}{\sin{\theta}+1}}$
