Solve: `(x+2y) (2x-y)`

Options

A. `2x^2+5xy+2y^2`
B. `2x^2+3xy-2y^2`
C. `x^2+4xy+y^2`
D. `x^2+4xy-y^2`

Answer

To solve the expression \((x + 2y)(2x - y)\), we will use the distributive property (also known as the FOIL method for binomials). Let's break it down step by step:

### Step 1: Distribute the first term
Multiply the first term of the first binomial by both terms of the second binomial:
\[ x \cdot 2x = 2x^2 \]

### Step 2: Distribute the second term
Now, multiply the first term of the first binomial by the second term of the second binomial:
\[ x \cdot (-y) = -xy \]

### Step 3: Distribute the third term
Next, multiply the second term of the first binomial by the first term of the second binomial:
\[ 2y \cdot 2x = 4xy \]

### Step 4: Distribute the fourth term
Finally, multiply the second term of the first binomial by the second term of the second binomial:
\[ 2y \cdot (-y) = -2y^2 \]

### Step 5: Combine all the results
Now, we will combine all the terms we obtained:
\[ 2x^2 - xy + 4xy - 2y^2 \]

### Step 6: Simplify the expression
Combine like terms:
\[ 2x^2 + (4xy - xy) - 2y^2 = 2x^2 + 3xy - 2y^2 \]

### Final Result
Thus, the simplified expression is:
\[ 2x^2 + 3xy - 2y^2 \]

### Conclusion
The answer matches the second option provided in the question.

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