I'm studying the basics of renormalization, and I want to know the nature of renormalization.

In 10.3 p.441 of Weinberg's QFT books,it states that

In actual calculations it is simplest just to say that the loop terms $\Pi_{LOOP}^*(q^2)$ we must subtract a first-order polynomial in $q^2$ with coefficients chosen so that the difference satisfies Eqs. (10.3.17) and (10.3.18). As we shall see, this subtraction procedure incidentally cancels the infinities that arise form the momentum space integrals in $\Pi^*_{LOOP}$.

However, as this discussion should make clear， the renormalization of masses and fields has nothing to do with the presence of infinities, and would be necessary even in a theory in which all momentum space integrals were convergent.

Eq. 10.3.17:\(\Pi^*(-m^2)=0\)

Eq. 10.3.18:\(\bigg[\frac{d}{dq^2}\Pi^*(q^2)\bigg]_{q^2=-m^2}=0\)

If this is true, the first-order polynomial must be equal to zero. Nothing could be subtracted!

What is wrong? What can be subtracted from the loop terms $\Pi_{LOOP}^*(q^2)$ ?

And another question, what infinities are canceled in this procedure?