First, thank you very much for giving us the complete problem that has you stuck.
Second, do you have any difficulty at all in understanding
[math]\dfrac{x \ \ln(x + 3)}{x^2 + 1} = 0 \implies x \ \ln(x + 3) = 0.[/math]
Third, do you have the slightest doubt that
[math]x \ \ln(x + 3) = 0 \implies x = 0 \text { or } \ln(x + 3) = 0.[/math]
If you have a problem in understanding, I suspect it is in one of the next two steps, but how do we know for sure if YOU don’t tell us where you lose the train of reasoning.