Question

pregunta 1-3
which expression is equivalent to (-2x^2 + 5x + 7)?
((2x + 7)(1 - x))
((2x - 7)(x + 1))
(-(2x + 7)(1 - x))
(-(2x - 7)(x + 1))

Explanation:

Step 1: Expand each option

We will expand each of the given options and check which one matches \(-2x^2 + 5x + 7\).

Option 1: \((2x + 7)(1 - x)\)

Using the distributive property (FOIL method):
\[

$$\begin{align*} (2x + 7)(1 - x)&=2x\times1-2x\times x + 7\times1-7\times x\\ &=2x-2x^2 + 7 - 7x\\ &=-2x^2+(2x - 7x)+7\\ &=-2x^2-5x + 7 \end{align*}$$

\]
This is not equal to \(-2x^2 + 5x + 7\).

Option 2: \((2x - 7)(x + 1)\)

Using the distributive property (FOIL method):
\[

$$\begin{align*} (2x - 7)(x + 1)&=2x\times x+2x\times1-7\times x - 7\times1\\ &=2x^2+2x-7x - 7\\ &=2x^2-5x - 7 \end{align*}$$

\]
This is not equal to \(-2x^2 + 5x + 7\).

Option 3: \(-(2x + 7)(1 - x)\)

First, we know from Option 1 that \((2x + 7)(1 - x)=-2x^2-5x + 7\). Now multiply by \(-1\):
\[

$$\begin{align*} -(2x + 7)(1 - x)&=-1\times(-2x^2-5x + 7)\\ &=2x^2 + 5x - 7 \end{align*}$$

\]
This is not equal to \(-2x^2 + 5x + 7\).

Option 4: \(-(2x - 7)(x + 1)\)

First, we know from Option 2 that \((2x - 7)(x + 1)=2x^2-5x - 7\). Now multiply by \(-1\):
\[

$$\begin{align*} -(2x - 7)(x + 1)&=-1\times(2x^2-5x - 7)\\ &=-2x^2 + 5x + 7 \end{align*}$$

\]
This matches the given expression \(-2x^2 + 5x + 7\).

Answer:

\(-(2x - 7)(x + 1)\) (the fourth option)