Solve the differential equations specified using an integrating factor.

Solve the initial-value problem

dy/dt = 1/(t+1)y + 4t^2 + 4t, y(1) = 10

Already,

Already,

so dy/dt = 1/(t+1)y + 4t^2 + 4t, y(1) = 10

by using the integrating factor(IF)

IF= e^(integral of P(x))

P(x)= 1/(t+1), so IF=e^(1/(t+1))=e^(ln(t+1))=t+1

now that I have the integrating factor I multiply everything by t+1

dy/dt = 1/(t+1)y + 4t^2 + 4t

(t+1) dy/dt = (t+1)(1/(t+1))y + (t+1)4t^2 + (t+1)4t

multiplying...

(t+1)dy/dt = y + 4t^3+4t^2 + 4t^2+4t

adding the ts...

(t+1) dy/dt = y + 4t^3 + 8t^2+4t, 4t^3 + 8t^2+4t= (t+1)(4t^2+4t)

dividing both sides by t+1

(t+1)/(t+1) dy/dt = (y + (t+1)(4t^2+4t))(t+1)

simplifying...

dy/dt= y/(t+1)+4t^2+4t

So far is the integrating by factors right?

And I'm getting stuck on dy/dt= y/(t+1)+4t^2+4t

Now how can I differentiate this first-order linear ordinary equation {dy/dt= y/(t+1)+4t^2+4t} ?