How do I know that this is the LCD (Lowest Common Denominator)

When presented with the following expression :

$\frac{1}{(x+1)(x+2)}-\frac{2}{x+1}+\frac{3}{x+2}$

I have used an online calculator to show me the LCD, which is :

$(x+2)(x+1)$

Although I can work out the LCD for different algebraic expressions, I am unsure of the methodology behind solving this one. I realise it must be basic but it just has not clicked yet.

If anybody could explain how this is done I would be very grateful.

When presented with the following expression :

$\frac{1}{(x+1)(x+2)}-\frac{2}{x+1}+\frac{3}{x+2}$

I have used an online calculator to show me the LCD, which is :

$(x+2)(x+1)$

Although I can work out the LCD for different algebraic expressions, I am unsure of the methodology behind solving this one. I realise it must be basic but it just has not clicked yet.

If anybody could explain how this is done I would be very grateful.