\(y' = \dfrac{y(t + 1) + (t + 1)^2}{t^2}\\ \dfrac{dy}{dt} = y\left(\dfrac1t + \dfrac1{t^2}\right) + \left(1 + \dfrac1t\right)^2\\ \dfrac{dy}{dt} - \left(\dfrac1t + \dfrac1{t^2}\right)y = \left(1 + \dfrac1t\right)^2\\ \)
We then find the integrating factor:
\(\mathcal{I}(t) = e^{-\int \left(\frac1t + \frac1{t^2}\right)\,dt} = e^{-\ln t + \frac1t} = \dfrac1t e^{1/t}\)
This means
\((y\mathcal I)' = \left(1 + \dfrac1t\right)^2 \mathcal I\)
\(\dfrac{y}t e^{1/t} = \displaystyle \int \dfrac1t \left(1 + \dfrac1t\right)^2 e^{1/t}\,dt\)
You will need some special functions to evaluate this integral, so good luck. Hope this helps.