Question about The Distributive Property I'm working through the basic math course on brilliant.org. I got the answer correct to the question (here's a link to the question), but I don't understand the solution for the question. 
Here are the solutions they provide. I'm learning, and what I don't understand is the how the right half of the top line in each example is related to the left half, and how the bottom line of each example is related to the top line. Can someone explain for me the logic of these examples below? 
For each example, on the top line I guess they're just splitting up the left side on the right side? I can see in example 1 how 8x(x) is the same as 8x, then they just moved the 3 outside the bracket and applied the 8x I guess. But then why is 8x(x) equivalent to 8x^2 ?  And why is there still an x after the 24? 8x(3) = 24 so shouldn't we just lose the x there as the calculation has been done?
Example 1
8x(x+3) = 8x(x) + 8x(3) 
        = 8x^2 + 24x

Example 2
4x(2x + 6) = 4x(2x) + 4x(6)
           = 8x^2 + 24x

Example 3
2x(4x+12) = 2x(4x) + 2x(12)
          = 8x^2 + 24x

Example 4
x^2(8+24) = x^2(8) + x^2(24)
          = 32x^2

 A: Short answer: You were assuming that some "x" symbols were the multiplication symbol $\times$. In fact, each is the variable $x$.
Long answer: The symbol "x" is confusing because it can stand for two things: multiplication $\times$ and the variable $x$. 
In more advanced math courses, people tend not to* use the $\times$ symbol for multiplication. Instead, they use one of two methods. First, the dot $\cdot$. So, you'd write $2\cdot 3 = 6$ instead of $ 2 \times 3 = 6$. Second, using parentheses. So, you'd write $2(3) = 6$, $(2)3 = 6$ or $(2)(3) = 6$ (all are equivalent) instead of $2 \times 3 = 6$.
This notation is useful when you work with the variable $x$. Note that $2x \cdot 3 = 6x$ and $(2x)3$ are much more legible than $2x \times 3 = 6x$, especially when you're writing these expressions by hand. Personally, I recommend using $\cdot$ or parentheses for multiplication whenever you're doing algebra with the variable $x$. It will help you avoid confusing $\times$ and $x$. 
Now, back to your question. Let's look at example 1. You want to expand the expression 8x(x+3) using the distributive property. You were interpreting this as $8\times (x + 3)$, so the first "x" was multiplication $\times$ and the second "x" was the variable $x$. But actually, the question is asking about $8x(x+3) = 8x \cdot (x+3)$. In these given examples, none of the "x" symbols stand for multiplication. They're all variables.
To solve the problem, you just split up the left and right side, exactly as you said. Here's the solution you supplied in your question, rewritten for clarity using the $\cdot$ notation:
\begin{align*}
8x \cdot (x+3) &= 8x \cdot x + 8x \cdot 3 \\
&= 8x^2 + 24x
\end{align*}
Does this help?
*Footnote so nobody yells at me: The $\times$ symbol actually is used in advanced math for special purposes. For example, the cross product (product of two vectors) and Cartesian product (product of two sets) use the symbol. Since vectors and sets rarely use the symbol $x$, using the $\times$ symbol is ok here.
