Factorize 12x2 - 24x + 9. 2. Find C such that x2 - 2x + C is a perfect square trinomial.

For a quadratic expression $ax^2 + bx + c$, we have $a=12$, $b=-24$, $c=9$.
The product $ac = 12 \times 9 = 108$.
The sum $b = -24$.

The two numbers are $-18$ and $-6$.
$(-18) \times (-6) = 108$
$(-18) + (-6) = -24$

Group the first two terms and the last two terms:

Factor out the greatest common factor from each group:

We can factor out a common factor of 3 from the first binomial:

The fully factored form is $\boxed{3(2x - 1)(2x - 3)}$.

2. Find $C$ such that $x^2 - 2x + C$ is a perfect square trinomial, and write the trinomial.

For a quadratic expression $ax^2 + bx + c$ to be a perfect square trinomial, $c = \left(\frac{b}{2a}\right)^2$.
For $x^2 - 2x + C$, we have $a=1$ and $b=-2$.

The perfect square trinomial is $x^2 - 2x + 1$.
This can be factored as $(x - 1)^2$.
The value of $C$ is $\boxed{1}$, and the trinomial is $\boxed{x^2 - 2x + 1}$.

3. Find $C$ such that $x^2 + 8x + C$ is a perfect square trinomial, and write the trinomial.

For $x^2 + 8x + C$, we have $a=1$ and $b=8$.

The perfect square trinomial is $x^2 + 8x + 16$.
This can be factored as $(x + 4)^2$.
The value of $C$ is $\boxed{16}$, and the trinomial is $\boxed{x^2 + 8x + 16}$.