Sure, let's analyze and simplify the given expression step by step.
We start with the expression:
[tex]\[
\frac{p + 2}{p^2 - p + 13}
\][/tex]
### Step-by-Step Solution:
1. Identify the components of the expression:
- The numerator is [tex]\( p + 2 \)[/tex].
- The denominator is [tex]\( p^2 - p + 13 \)[/tex].
2. Check for factorization:
- We need to check if the numerator or the denominator can be factored further.
- The numerator, [tex]\( p + 2 \)[/tex], is already in its simplest form.
- The quadratic expression in the denominator, [tex]\( p^2 - p + 13 \)[/tex], does not factor nicely over the real numbers (it does not have real roots because the discriminant [tex]\( (b^2 - 4ac) = (-1)^2 - 4(1)(13) = 1 - 52 = -51 \)[/tex] is negative).
3. Simplifying the expression:
- Since neither the numerator nor the denominator can be factored further or simplified, the expression remains in its simplest form.
Thus, the simplified form of the given expression is:
[tex]\[
\boxed{\frac{p + 2}{p^2 - p + 13}}
\][/tex]
Since there are no common factors to cancel between the numerator and denominator, this is the final, simplified version of the expression.
