Area of a rectangular field is `(2x^(3)-11x^(2)-4x+5)`sq. units and side of a square field is `(2x^(2)+4)` units. Find the difference between their areas (in sq. units)

Options

A. `4x^(4)-2x^(3)-27x^(2)-4x+11`
B. `4x^(4)-2x^(3)+27x^(2)+4x+11`
C. `4x^(4)+27x^(2)+4x-11`
D. `4x^(4)+2x^(3)+27x^(2)+4x+11`

Answer

To find the difference between the areas of a rectangular field and a square field, we will follow these steps:

### Step 1: Calculate the Area of the Square Field
The area of a square is given by the formula:
\[ \text{Area} = (\text{side})^2 \]
Here, the side of the square field is \(2x^2 + 4\). Therefore, we calculate:
\[ \text{Area of square} = (2x^2 + 4)^2 \]
Using the formula for the square of a binomial \((a + b)^2 = a^2 + 2ab + b^2\), we have:
- \(a = 2x^2\)
- \(b = 4\)

Calculating each term:
\[ a^2 = (2x^2)^2 = 4x^4 \]
\[ b^2 = 4^2 = 16 \]
\[ 2ab = 2 \cdot (2x^2) \cdot 4 = 16x^2 \]

Putting it all together:
\[ \text{Area of square} = 4x^4 + 16x^2 + 16 \]

### Step 2: Write Down the Area of the Rectangular Field
The area of the rectangular field is given as:
\[ \text{Area of rectangle} = 2x^3 - 11x^2 - 4x + 5 \]

### Step 3: Find the Difference Between the Areas
To find the difference between the area of the square and the area of the rectangle, we subtract the area of the rectangle from the area of the square:
\[ \text{Difference} = \text{Area of square} - \text{Area of rectangle} \]
Substituting the areas we calculated:
\[ \text{Difference} = (4x^4 + 16x^2 + 16) - (2x^3 - 11x^2 - 4x + 5) \]

### Step 4: Simplify the Expression
Distributing the negative sign:
\[ \text{Difference} = 4x^4 + 16x^2 + 16 - 2x^3 + 11x^2 + 4x - 5 \]

Now, combine like terms:
- The \(x^4\) term: \(4x^4\)
- The \(x^3\) term: \(-2x^3\)
- The \(x^2\) terms: \(16x^2 + 11x^2 = 27x^2\)
- The \(x\) term: \(4x\)
- The constant terms: \(16 - 5 = 11\)

Putting it all together, we get:
\[ \text{Difference} = 4x^4 - 2x^3 + 27x^2 + 4x + 11 \]

### Final Answer
The difference between the areas of the rectangular field and the square field is:
\[ \boxed{4x^4 - 2x^3 + 27x^2 + 4x + 11} \]