The expression shown in the image is:

$\frac{1}{x - x^2} + 1$

To simplify or analyze this expression, let’s proceed with the following steps:

Step 1: Simplify the Fraction

Rewrite the expression with a common denominator if possible. We can transform $+1$ to have the same denominator as $\frac{1}{x - x^2}$.

Step 2: Find the Common Denominator

Rewrite $1$ as $\frac{x - x^2}{x - x^2}$: $\frac{1}{x - x^2} + \frac{x - x^2}{x - x^2}$

Step 3: Combine Terms

Now, combining the fractions: $\frac{1 + (x - x^2)}{x - x^2} = \frac{1 + x - x^2}{x - x^2}$

Step 4: Simplify the Numerator

The numerator simplifies to: $1 + x - x^2$

So, the expression becomes: $\frac{1 + x - x^2}{x - x^2}$

Let me know if you'd like a detailed breakdown or have any questions!

Here are 5 questions to expand on this topic:

1. How can we simplify expressions involving fractions and addition?
2. What are some strategies to factor quadratic expressions?
3. How does simplifying rational expressions help in solving equations?
4. What are common techniques for combining terms with different denominators?
5. How can we analyze the domain of rational expressions like this one?

Tip: Always look for opportunities to factor expressions, as it can reveal ways to simplify or cancel terms!