Question

What is the quotient?

Answer 1

$\text{First factor}\ t^2+7t+12=t^2+4t+3t+12=t(t+4)+3(t+4)\\\\=t^2+7t+12=(t+4)(t+3)\\\\\dfrac{t+3}{t+4}\div(t^2+7t+12)=\dfrac{t+3}{t+4}\div[(t+4)(t+3)]\\\\=\dfrac{t+3}{t+4}\cdot\dfrac{1}{(t+4)(t+3)}\\\\\text{canceled}\ t+3\\\\=\dfrac{1}{t+4}\cdot\dfrac{1}{t+4}=\boxed{\dfrac{1}{(t+4)^2}}$