The question in your image is asking you to complete the expressions so they form "trinomios cuadrados perfectos" (perfect square trinomials). To solve this, we follow a specific rule for perfect square trinomials:

For an expression like $x^2 + bx + c$ to be a perfect square trinomial, the value of $c$ must be $\left( \frac{b}{2} \right)^2$.

Let's solve each expression from the image:

a. $x^2 + \_ + 49$

For this to be a perfect square, the middle term should be $2 \times x \times 7 = 14x$, so the missing term is 14x.

b. $x^2 - \_ + 1$

For this to be a perfect square, the middle term should be $2 \times x \times 1 = 2x$, so the missing term is -2x.

c. $x^2 + 10x + \_$

Here $b = 10$. To complete the square: $\left( \frac{10}{2} \right)^2 = 25$ So the missing term is 25.

d. $x^2 + \_ + 36$

For this to be a perfect square, the middle term should be $2 \times x \times 6 = 12x$, so the missing term is 12x.

e. $x^2 - 6x + \_$

Here $b = -6$. To complete the square: $\left( \frac{-6}{2} \right)^2 = 9$ So the missing term is 9.

f. $x^2 - 16x + \_$

Here $b = -16$. To complete the square: $\left( \frac{-16}{2} \right)^2 = 64$ So the missing term is 64.

g. $x^2 + 18x + \_$

Here $b = 18$. To complete the square: $\left( \frac{18}{2} \right)^2 = 81$ So the missing term is 81.

h. $x^2 + \_ + 121$

For this to be a perfect square, the middle term should be $2 \times x \times 11 = 22x$, so the missing term is 22x.

Would you like any further explanation on how perfect square trinomials work or how to complete the square?

Here are 5 related questions to practice:

1. How do you factor a perfect square trinomial?
2. What is the general formula for completing the square?
3. How would you solve $x^2 + 12x + 36 = 0$?
4. Can a trinomial with a negative constant be a perfect square?
5. How can completing the square help solve quadratic equations?

Tip: Completing the square is a method often used to derive the quadratic formula!