Step-by-step explanation:
So you don't understand the way the answer is written? Either way, I'll explain how to get the domain and how to interpret the answer.
So the domain is what values can we plug in for x. This has the potential to be tricky. basically though you need to look at an equation and think of what values won't give an answer. The two easiest to explain are 1/x or 
. You can't divide by 0 and you can't take a square root of a negative number. so for 1/x, that x can't equal 0 and for the square root of x, it has to be greater than or equal to 0.
You can write these conditions in the form shown as an answer to the question. For every number but 0 you would write (-∞,0)∪(0,∞) and greater than or equal to 0 is [0,∞) and this has a few parts to interpret.
First, parenthesis or brackets. If you have parenthesis it means up to this number but not including it. Or in other words just greatee than or less than, and not equal to. so < or >. brackets are the opposite, where it does include them so with the domain of a square root, it can be 0 so it's included so we use a bracket. But for a rational function 1/x the denominator can't equal 0 so it can be everything up to 0, but not including it, and then everything from 0 but not including it. Also, if it ever goes to positive or negative infinity it also uses parentheses.
The next part that might need explanations are those ∪s. They are the symbol for union, which is from set notation. Basically what union does is it is placed between two "sets" and it tells you you are going to use everything from the left and right set. So for the domain of a rational function (-∞,0)∪(0,∞) you would think of it as negative infinity to 0, but not including it AND 0 bu not including it to infinity. That AND is what the union symbol represents.
Anyway, now onto your function. It is a rational function, so you need to make sure the denominator doesn't equal zero. Well to know when it equals 0 you sole for when it would. Let me know if you need help with this. It gives you the answer though. the domain is (-∞,-4)∪(-4,-1)∪(-1,∞). You could understand this as the domain is negative infinity to -4 but not including it to -1 but not including it to infinity, I didn't include the ands representing the union symbol, but hopefully you see where they would go.
 = \frac{8x + 12}{ {x}^{2} + 5x + 4 } )
 \: u \: ( - 4, - 1) \: u \: ( - 1, \infty ))
