To add the two fractions given and simplify the answer, let's go through the steps carefully.

The expression to add is:

$\frac{x + 5}{2x + 2} + \frac{x}{1 - x^2}$

Step 1: Factor the Denominators

1. In the first fraction, $2x + 2$ can be factored as $2(x + 1)$.
2. In the second fraction, $1 - x^2$ can be factored using the difference of squares as $(1 - x)(1 + x)$.

So, we rewrite the expression as:

$\frac{x + 5}{2(x + 1)} + \frac{x}{(1 - x)(x + 1)}$

Step 2: Find a Common Denominator

The common denominator of $2(x + 1)$ and $(1 - x)(x + 1)$ is $2(x + 1)(1 - x)$.

Step 3: Rewrite Each Fraction with the Common Denominator

1. For the first fraction, multiply the numerator and the denominator by $1 - x$: $\frac{(x + 5)(1 - x)}{2(x + 1)(1 - x)}$

2. For the second fraction, multiply the numerator and the denominator by $2$: $\frac{2x}{2(x + 1)(1 - x)}$

Now we have:

$\frac{(x + 5)(1 - x) + 2x}{2(x + 1)(1 - x)}$

Step 4: Expand and Simplify the Numerator

Expand $(x + 5)(1 - x)$:

$(x + 5)(1 - x) = x - x^2 + 5 - 5x = -x^2 - 4x + 5$

Now, add $-x^2 - 4x + 5$ and $2x$:

$-x^2 - 4x + 5 + 2x = -x^2 - 2x + 5$

Final Answer

Thus, the simplified form of the expression is:

$\frac{-x^2 - 2x + 5}{2(x + 1)(1 - x)}$

This matches the form given in the question.

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Would you like further details on any specific step?

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Relative Questions:

1. How do you find a common denominator for complex rational expressions?
2. What other methods can you use to simplify complex fractions?
3. How can factoring help in simplifying rational expressions?
4. What is the importance of simplifying expressions in algebra?
5. How does the difference of squares factorization work?

Tip: Always start by factoring the denominators when working with rational expressions; it helps in finding a common denominator easily.