(2y)/(x ^ 2 + xy) - (2y)/(2x ^ 2 + 3xy + y ^ 2) + 1/x

We are tasked with simplifying the following expression:

\[
\frac{2y}{x^2 + xy} - \frac{2y}{2x^2 + 3xy + y^2} + \frac{1}{x}
\]

### Step 1: Factor the denominators
1. **First term:**
   \[
   \frac{2y}{x^2 + xy} = \frac{2y}{x(x + y)}
   \]
   Here, we factored out \(x\) from the denominator.

2. **Second term:**
   \[
   \frac{2y}{2x^2 + 3xy + y^2}
   \]
   This is a quadratic expression. We check for factorizations:
   \[
   2x^2 + 3xy + y^2 = (2x + y)(x + y)
   \]
   So, the second term becomes:
   \[
   \frac{2y}{(2x + y)(x + y)}
   \]

### Step 2: Combine the terms
Now we have:
\[
\frac{2y}{x(x + y)} - \frac{2y}{(2x + y)(x + y)} + \frac{1}{x}
\]

To combine these terms, the common denominator will be \(x(x + y)(2x + y)\). Let’s rewrite each fraction with this common denominator:

1. **First term:**
   \[
   \frac{2y}{x(x + y)} = \frac{2y(2x + y)}{x(x + y)(2x + y)}
   \]

2. **Second term:**
   \[
   \frac{2y}{(2x + y)(x + y)} = \frac{2yx}{x(x + y)(2x + y)}
   \]

3. **Third term:**
   \[
   \frac{1}{x} = \frac{(x + y)(2x + y)}{x(x + y)(2x + y)}
   \]

### Step 3: Combine the numerators
Now that we have a common denominator, we can combine the numerators:

\[
\frac{2y(2x + y)}{x(x + y)(2x + y)} - \frac{2yx}{x(x + y)(2x + y)} + \frac{(x + y)(2x + y)}{x(x + y)(2x + y)}
\]

The common denominator is \(x(x + y)(2x + y)\). So, combine the numerators:

\[
2y(2x + y) - 2yx + (x + y)(2x + y)
\]

### Step 4: Simplify the numerator
Now, expand and simplify the numerator:

1. \(2y(2x + y) = 4xy + 2y^2\)
2. \(-2yx = -2xy\)
3. \((x + y)(2x + y)\): Use distributive property here:
   \[
   (x + y)(2x + y) = x(2x + y) + y(2x + y) = 2x^2 + xy + 2xy + y^2 = 2x^2 + 3xy + y^2
   \]

Now combine all the terms:
\[
4xy + 2y^2 - 2xy + 2x^2 + 3xy + y^2 = 2x^2 + 5xy + 3y^2
\]

Thus, the simplified expression becomes:
\[
\frac{2x^2 + 5xy + 3y^2}{x(x + y)(2x + y)}
\]

This is the final simplified form of the given expression.

---

Do you need further details or have any questions?

Here are some follow-up questions:
1. What are the key differences between factoring and expanding?
2. How do common denominators work when adding rational expressions?
3. Can you apply the quadratic formula to the second denominator \(2x^2 + 3xy + y^2\)?
4. How does factoring simplify algebraic expressions in general?
5. What is the importance of recognizing common terms when simplifying?

**Tip:** Always factor expressions whenever possible before adding or subtracting fractions—it simplifies your calculations significantly.

(2y)/(x^2 + xy) - (2y)/(2x^2 + 3xy + y^2) + 1/x

This detailed step-by-step solution simplifies the expression (2y)/(x^2 + xy) - (2y)/(2x^2 + 3xy + y^2) + 1/x by factoring and combining terms. Learn key algebra concepts such as factoring and the common denominator method, suitable for Grades 9-12.

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