Let's break down the two problems presented in the image.

Problem 1

The problem asks to calculate the following summation:

$\sum_{x=1}^{12} x(x + 1)$

We need to evaluate this summation by expanding the expression $x(x + 1)$ for each $x$ from 1 to 12.

Problem 2

This one involves a different type of summation. Given the following values: $x_1 = 7, \, x_2 = 13, \, x_3 = 14, \, x_4 = 17, \, x_5 = 19, \, x_6 = 22$

We are asked to calculate:

$\sum_{i=1}^{4} (x_i - 2)$

For this, we will sum the values of $x_i - 2$ for the first four values of $x_i$ provided.

Let me solve these two problems now.### Solution to Problem 1 The value of the summation:

$\sum_{x=1}^{12} x(x + 1) = 728$

Solution to Problem 2

The value of the summation:

$\sum_{i=1}^{4} (x_i - 2) = 43$

Would you like more details on these solutions, or have any questions about the process?

Here are 5 related questions you can explore:

1. How would the result of Problem 1 change if the upper limit were extended to $x = 15$?
2. Can the expression $x(x + 1)$ be simplified using algebraic identities before summing?
3. How does changing the constant value $2$ in Problem 2 affect the summation result?
4. What is the general formula for the sum of $x(x + 1)$ from 1 to $n$?
5. How would you compute the sum of the squares of the values $x_i$?

Tip: Summation formulas like $\sum_{x=1}^{n} x$ or $\sum_{x=1}^{n} x^2$ can be powerful tools for quickly calculating large sums.