Let's work through the two questions one by one.

Question 1: Which value must be added to the expression $x^2 - 8x$ to make it a perfect-square trinomial?

To make a quadratic expression $x^2 + bx$ into a perfect-square trinomial, the value to be added is $\left(\frac{b}{2}\right)^2$.

For the expression $x^2 - 8x$, the value of $b$ is $-8$.

1. First, divide $b$ by 2:
   $\frac{-8}{2} = -4$
2. Square the result:
   $(-4)^2 = 16$

Thus, 16 must be added to the expression to make it a perfect-square trinomial.
The correct answer is 16.

Question 2: Which of the following expressions is a perfect square trinomial?

A perfect square trinomial has the form $(x + a)^2 = x^2 + 2ax + a^2$.

Let's check each option:

1. $x^2 + 4x + 8$:
   The middle term is 4, so $a$ should be $\frac{4}{2} = 2$.
   But $a^2 = 2^2 = 4$, not 8. Therefore, this is not a perfect square trinomial.

2. $x^2 + 8x + 15$:
   The middle term is 8, so $a$ should be $\frac{8}{2} = 4$.
   But $a^2 = 4^2 = 16$, not 15. This is not a perfect square trinomial.

3. $x^2 - 10x + 25$:
   The middle term is -10, so $a$ should be $\frac{-10}{2} = -5$.
   $a^2 = (-5)^2 = 25$, which matches the constant term.
   Thus, this is a perfect square trinomial.

4. $x^2 + 6x + 12$:
   The middle term is 6, so $a$ should be $\frac{6}{2} = 3$.
   But $a^2 = 3^2 = 9$, not 12. This is not a perfect square trinomial.

The correct answer is $x^2 - 10x + 25$.

Would you like further explanation on completing the square or identifying perfect-square trinomials?

Here are some additional questions for further practice:

1. How do you complete the square for a quadratic equation?
2. Can a quadratic expression with a negative constant term be a perfect square?
3. What is the general formula for factoring perfect-square trinomials?
4. How would you solve a quadratic equation using the method of completing the square?
5. What is the significance of the discriminant in determining whether a quadratic is factorable?

Tip: A quick way to recognize a perfect-square trinomial is to check if the constant term is the square of half the coefficient of the linear term.