Factor polynomial $$ x^{2}+2x+3 $$

It seems that $ x^{2}+2x+3 $ cannot be factored out.

Step 1: Identify constants $ \color{blue}{ b }$ and $\color{red}{ c }$. ( $ \color{blue}{ b }$ is a number in front of the $ x $ term and $ \color{red}{ c } $ is a constant). In our case:

Now we must discover two numbers that sum up to $ \color{blue}{ 2 } $ and multiply to $ \color{red}{ 3 } $.

Step 2: Find out pairs of numbers with a product of $\color{red}{ c = 3 }$.

Step 3: Because none of these pairs will give us a sum of $ \color{blue}{ 2 }$, we conclude the polynomial cannot be factored.