Find the coefficient of `x^3y in (x + 2y) xx (5x + y)^3`

Options

A. 250
B. 175
C. 475
D. 325

Answer

To find the coefficient of \( x^3y \) in the expression \( (x + 2y)(5x + y)^3 \), we will follow these steps:

### Step 1: Expand \( (5x + y)^3 \)
We will use the binomial expansion formula:

\[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]

In our case, \( a = 5x \), \( b = y \), and \( n = 3 \).

\[ (5x + y)^3 = \sum_{k=0}^{3} \binom{3}{k} (5x)^{3-k} y^k \]

Calculating the terms:

- For \( k = 0 \): \( \binom{3}{0} (5x)^3 y^0 = 1 \cdot 125x^3 = 125x^3 \)
- For \( k = 1 \): \( \binom{3}{1} (5x)^2 y^1 = 3 \cdot 25x^2 \cdot y = 75x^2y \)
- For \( k = 2 \): \( \binom{3}{2} (5x)^1 y^2 = 3 \cdot 5x \cdot y^2 = 15xy^2 \)
- For \( k = 3 \): \( \binom{3}{3} (5x)^0 y^3 = 1 \cdot y^3 = y^3 \)

Putting it all together:

\[ (5x + y)^3 = 125x^3 + 75x^2y + 15xy^2 + y^3 \]

### Step 2: Multiply by \( (x + 2y) \)
Now we multiply \( (x + 2y) \) with the expanded form:

\[ (x + 2y)(125x^3 + 75x^2y + 15xy^2 + y^3) \]

Distributing \( x \) and \( 2y \):

1. \( x \cdot 125x^3 = 125x^4 \)
2. \( x \cdot 75x^2y = 75x^3y \)
3. \( x \cdot 15xy^2 = 15x^2y^2 \)
4. \( x \cdot y^3 = xy^3 \)
5. \( 2y \cdot 125x^3 = 250x^3y \)
6. \( 2y \cdot 75x^2y = 150x^2y^2 \)
7. \( 2y \cdot 15xy^2 = 30xy^3 \)
8. \( 2y \cdot y^3 = 2y^4 \)

Combining all these terms, we have:

\[ 125x^4 + (75x^3y + 250x^3y) + (15x^2y^2 + 150x^2y^2) + (xy^3 + 30xy^3) + 2y^4 \]

This simplifies to:

\[ 125x^4 + 325x^3y + 165x^2y^2 + 31xy^3 + 2y^4 \]

### Step 3: Identify the coefficient of \( x^3y \)
From the combined expression, we see that the coefficient of \( x^3y \) is \( 325 \).

### Final Answer
The coefficient of \( x^3y \) in the expression \( (x + 2y)(5x + y)^3 \) is \( \boxed{325} \).

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