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$$ \frac{\left(x+3\right)}{x\left(x+1\right)}-\frac{\left(x+4\right)}{x^2+2x+1} $$

To simplify the expression 

\[
\frac{x + 3}{x(x + 1)} - \frac{x + 4}{x^2 + 2x + 1},
\]

we start by noticing that $x^2 + 2x + 1$ can be factored as $(x + 1)^2$. Thus, the expression becomes:

\[
\frac{x + 3}{x(x + 1)} - \frac{x + 4}{(x + 1)^2}.
\]

Next, we find a common denominator for both fractions. The common denominator is $x(x + 1)^2$. We can rewrite each fraction:

1. For the first fraction:
   \[
   \frac{x + 3}{x(x + 1)} = \frac{(x + 3)(x + 1)}{x(x + 1)^2}.
   \]

2. For the second fraction:
   \[
   \frac{x + 4}{(x + 1)^2} = \frac{(x + 4)x}{x(x + 1)^2}.
   \]

Now we can combine the fractions:

\[
\frac{(x + 3)(x + 1) - (x + 4)x}{x(x + 1)^2}.
\]

Next, we expand the numerators:

1. Expanding $(x + 3)(x + 1)$:
   \[
   (x + 3)(x + 1) = x^2 + 4x + 3.
   \]

2. Expanding $(x + 4)x$:
   \[
   (x + 4)x = x^2 + 4x.
   \]

Now substituting back into our combined fraction:

\[
\frac{x^2 + 4x + 3 - (x^2 + 4x)}{x(x + 1)^2}.
\]

Simplifying the numerator:

\[
x^2 + 4x + 3 - x^2 - 4x = 3.
\]

Thus, we have:

\[
\frac{3}{x(x + 1)^2}.
\]

The final simplified expression is:

\[
\frac{3}{x(x + 1)^2}.
\]

If you have any questions or would like further details, let me know! Here are some related questions to consider:

1. What happens to the expression if $x = 0$?
2. How can we find the domain of the simplified expression?
3. What is the behavior of the function as $x$ approaches infinity?
4. How can you verify the simplification step-by-step?
5. What are the implications if $x + 1 = 0$?

**Tip:** Always check for restrictions in the domain when dealing with rational expressions!

förenkla $ \frac{\left(x+3\right)}{x\left(x+1\right)}-\frac{\left(x+4\right)}{x^2+2x+1} $

Discover how to simplify the rational expression $ \frac{\left(x+3\right)}{x\left(x+1\right)} - \frac{\left(x+4\right)}{(x+1)^2} $ step-by-step. This solution involves factoring and finding common denominators, making it suitable for students in Grades 9-11.

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