Perform the indicated operation and simplify.
StartFraction 5 x plus 4 y Over x squared plus 4 xy plus 4 y squared EndFraction
timesStartFraction x plus 2 y Over 2 EndFraction

To solve the expression

\[
\frac{5x + 4y}{x^2 + 4xy + 4y^2} \times \frac{x + 2y}{2},
\]

we need to multiply the two fractions and then simplify the resulting expression. Here’s the step-by-step process:

### Step 1: Factor the denominator of the first fraction
Notice that the denominator of the first fraction, \(x^2 + 4xy + 4y^2\), is a perfect square trinomial. It can be factored as:

\[
x^2 + 4xy + 4y^2 = (x + 2y)^2.
\]

Thus, the expression becomes:

\[
\frac{5x + 4y}{(x + 2y)^2} \times \frac{x + 2y}{2}.
\]

### Step 2: Multiply the fractions
Multiply the numerators and the denominators:

\[
\frac{(5x + 4y) \times (x + 2y)}{(x + 2y)^2 \times 2}.
\]

### Step 3: Simplify the expression
The numerator has a factor of \(x + 2y\), which is also present in the denominator:

\[
\frac{(5x + 4y) \times (x + 2y)}{2 \times (x + 2y)^2}.
\]

Cancel out one of the \(x + 2y\) factors from the numerator and denominator:

\[
\frac{5x + 4y}{2 \times (x + 2y)}.
\]

### Final Answer:
The simplified form of the expression is:

\[
\frac{5x + 4y}{2(x + 2y)}.
\]

Would you like any further details or have any questions?

Here are some related questions to expand your understanding:
1. How do you identify and factor a perfect square trinomial?
2. What are the steps to simplify a fraction when there are common factors in the numerator and denominator?
3. Can the expression \(5x + 4y\) be factored further?
4. Why is it important to cancel out common factors when simplifying rational expressions?
5. How do you multiply and simplify algebraic fractions in general?

**Tip:** Always look for common factors between the numerator and denominator to simplify expressions efficiently.

Learn how to simplify the expression (5x + 4y) / (x^2 + 4xy + 4y^2) * (x + 2y) / 2 step-by-step. This problem involves algebraic fraction simplification and factorization of perfect square trinomials. Suitable for high school students.

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