Expand and simplify `(2x + 5) ^(2)(4x +1) - (3x ^(3) - 16 x ^(2) + 25 x + 21)`

Options

A. `13 x ^(3) + 92x ^(2) + 55x - 45`
B. `13 x ^(3) - 92 x ^(2) - 55 x -4`
C. `12 x ^(3) - 92 x ^(2) + 55x - 4`
D. `13 x ^(2) - 92 x ^(2) + 55 x + 4`

Answer

To solve the expression \((2x + 5)^{2}(4x + 1) - (3x^{3} - 16x^{2} + 25x + 21)\), we will follow these steps:

### Step 1: Expand \((2x + 5)^{2}\)

Using the formula for the square of a binomial, \((a + b)^{2} = a^{2} + 2ab + b^{2}\):

- Here, \(a = 2x\) and \(b = 5\).
- So, \((2x + 5)^{2} = (2x)^{2} + 2(2x)(5) + (5)^{2}\).

Calculating each term:
- \((2x)^{2} = 4x^{2}\)
- \(2(2x)(5) = 20x\)
- \((5)^{2} = 25\)

Thus, \((2x + 5)^{2} = 4x^{2} + 20x + 25\).

### Step 2: Multiply the result by \((4x + 1)\)

Now we need to multiply \((4x + 1)\) with \((4x^{2} + 20x + 25)\):

\[ (4x + 1)(4x^{2} + 20x + 25) \]

Distributing \(4x\):
- \(4x \cdot 4x^{2} = 16x^{3}\)
- \(4x \cdot 20x = 80x^{2}\)
- \(4x \cdot 25 = 100x\)

Distributing \(1\):
- \(1 \cdot 4x^{2} = 4x^{2}\)
- \(1 \cdot 20x = 20x\)
- \(1 \cdot 25 = 25\)

Combining these results:
\[ 16x^{3} + (80x^{2} + 4x^{2}) + (100x + 20x) + 25 = 16x^{3} + 84x^{2} + 120x + 25 \]

### Step 3: Subtract the second part \((3x^{3} - 16x^{2} + 25x + 21)\)

Now we need to subtract the expression:
\[ (16x^{3} + 84x^{2} + 120x + 25) - (3x^{3} - 16x^{2} + 25x + 21) \]

Distributing the negative sign:
\[ = 16x^{3} + 84x^{2} + 120x + 25 - 3x^{3} + 16x^{2} - 25x - 21 \]

Combining like terms:
- For \(x^{3}\): \(16x^{3} - 3x^{3} = 13x^{3}\)
- For \(x^{2}\): \(84x^{2} + 16x^{2} = 100x^{2}\)
- For \(x\): \(120x - 25x = 95x\)
- For the constant: \(25 - 21 = 4\)

Thus, the final result is:
\[ 13x^{3} + 100x^{2} + 95x + 4 \]

### Final Answer:
\[ 13x^{3} + 100x^{2} + 95x + 4 \]

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