If `f(x)=3-2x+x^(2)`, then `((f(x+t)-f(x))/(t))`=

Options

A. `t^(2)+2xt-2t`
B. `x^(2)t^(2)-2xt+3`
C. `t+2x-2`
D. `2x-2`

Explanation

To solve the problem, we need to find the expression \(\frac{f(x+t) - f(x)}{t}\) where \(f(x) = 3 - 2x + x^2\).

### Step-by-Step Solution:

1. **Substitute \(x+t\) into \(f(x)\)**:
\[ f(x+t) = 3 - 2(x+t) + (x+t)^2 \]
Expanding this:
\[ f(x+t) = 3 - 2x - 2t + (x^2 + 2xt + t^2) \]
Combining like terms:
\[ f(x+t) = 3 - 2x - 2t + x^2 + 2xt + t^2 \]

2. **Write down \(f(x)\)**:
\[ f(x) = 3 - 2x + x^2 \]

3. **Calculate \(f(x+t) - f(x)\)**:
\[ f(x+t) - f(x) = (3 - 2x - 2t + x^2 + 2xt + t^2) - (3 - 2x + x^2) \]
Simplifying this:
\[ f(x+t) - f(x) = -2t + 2xt + t^2 \]

4. **Divide by \(t\)**:
\[ \frac{f(x+t) - f(x)}{t} = \frac{-2t + 2xt + t^2}{t} \]
Breaking this down:
\[ = \frac{-2t}{t} + \frac{2xt}{t} + \frac{t^2}{t} \]
Which simplifies to:
\[ = -2 + 2x + t \]

5. **Final Expression**:
Thus, we have:
\[ \frac{f(x+t) - f(x)}{t} = t + 2x - 2 \]

### Final Answer:
\[ \frac{f(x+t) - f(x)}{t} = t + 2x - 2 \]