relations between Minimal Polynomial on Quotient Space, invariant subspace and original vector space

Question

I just see a question, asks if:

V is a vector space with T as a linear transformation. W is a T-invariant subspace of V. Then we have V/W as a quotient.

So it asks me to judge possible minimal polynomial of T on V while given:

1. p_mini_W = (x-1) and p_quotient = x^2+1.

In this case, since we know both should divide the minimal polynomial in V, it should have the form f(x)(x-1)(x^2+1)

But another case is:

2. p_mini_W = (x-1) and p_quotient = (x-1)^2.

In this case, what is the difference? does form f(x)*(x-1)^2 also works? Or do I missed something important?