Answer
The trinomial\[{{x}^{2}}+2x+3\] is prime.
Work Step by Step
Note that this trinomial has leading coefficient is 1, therefore to factor it follow the steps below:
Step 1:
Find first two terms whose product is\[{{x}^{2}}\].
\[{{x}^{2}}+2x+3=\left( x\text{ } \right)\left( x\text{ } \right)\]
Step 2:
Find the last two terms which when multiplied gives \[3\]:
\[\text{Factors of }3:\text{ }-1,-3\text{ }1,3\]
Step 3:
Now try various combinations of these factors and look for the pair of factors whose sum is the coefficient of the middle term. That is, \[2\]in this case.
\[\begin{align}
& \text{Factors of }3:\text{ }-1,-3\text{ }1,3 \\
& \text{Sum of factors : }-4\text{ }4\text{ } \\
\end{align}\]
Since, none of the possible factors yields the sum \[2\]. This implies that the given trinomial cannot be factored using the integers and therefore, the trinomial\[{{x}^{2}}+2x+3\] is prime.
