Why $\Theta(n^2)$ multiplication of coefficient required for canonical form of polynomial?
Multiplying a degree $d$ polynomial by a degree $1$ monic polynomial requires $d$ multiplications (and some additions). This can be seen from the following formula:
$$
(x-c) \sum_{i=0}^d b_i x^i = (-cb_0) x^0 + \sum_{i=1}^d (b_{i-1}-cb_i)x^i + b_dx^d.
$$
If you compute $(x-a_1)\cdots(x-a_n)$ by successively multiplying the terms, i.e. by computing $P_1 = x-a_1$ and $P_{d+1} = P_d(x-a_{d+1})$ for $1 \leq d \leq n-1$, then you need this many multiplications:
$$
1+2+\cdots+(n-1) = \frac{n(n-1)}{2} = \Theta(n^2).
$$