For all x, `(2x +5)^2(-3x + 7)= ?`

Options

A. `-12x^2 + 70`
B. `-6x^2-x +35`
C. `4x^2+ 20x + 25`
D. `-12x^3 - 32x^2 + 65x + 175`

Answer

To solve the expression \((2x + 5)^2(-3x + 7)\), we will follow these steps:

### Step 1: Expand \((2x + 5)^2\)
Using the formula for the square of a binomial, \((a + b)^2 = a^2 + 2ab + b^2\):
\[ (2x + 5)^2 = (2x)^2 + 2(2x)(5) + (5)^2 \]
Calculating each term:
\[ (2x)^2 = 4x^2 \]
\[ 2(2x)(5) = 20x \]
\[ (5)^2 = 25 \]
So,
\[ (2x + 5)^2 = 4x^2 + 20x + 25 \]

### Step 2: Multiply the expanded form by \(-3x + 7\)
Now we will multiply \((4x^2 + 20x + 25)\) by \((-3x + 7)\):
\[ (4x^2 + 20x + 25)(-3x + 7) \]
We will distribute each term in the first polynomial by each term in the second polynomial.

#### Distributing \(4x^2\):
\[ 4x^2 \cdot (-3x) = -12x^3 \]
\[ 4x^2 \cdot 7 = 28x^2 \]

#### Distributing \(20x\):
\[ 20x \cdot (-3x) = -60x^2 \]
\[ 20x \cdot 7 = 140x \]

#### Distributing \(25\):
\[ 25 \cdot (-3x) = -75x \]
\[ 25 \cdot 7 = 175 \]

### Step 3: Combine all the terms
Now we will combine all the terms we obtained:
\[ -12x^3 + (28x^2 - 60x^2) + (140x - 75x) + 175 \]
This simplifies to:
\[ -12x^3 - 32x^2 + 65x + 175 \]

### Final Expression
Thus, the final expression is:
\[ (2x + 5)^2(-3x + 7) = -12x^3 - 32x^2 + 65x + 175 \]

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