Sure, let's work through the steps to solve this problem.
Given the function [tex]\( f(x) = -2x^2 - x - 3 \)[/tex], we want to find the expression for
[tex]\[ \frac{f(x+h) - f(x)}{h} \][/tex]
### Step 1: Compute [tex]\( f(x+h) \)[/tex]
We start by finding [tex]\( f(x+h) \)[/tex]. This means we substitute [tex]\( x+h \)[/tex] into the function [tex]\( f(x) \)[/tex]:
[tex]\[ f(x+h) = -2(x+h)^2 - (x+h) - 3 \][/tex]
### Step 2: Expand [tex]\( f(x+h) \)[/tex]
Now we expand the expression:
[tex]\[ f(x+h) = -2(x+h)^2 - (x+h) - 3 \][/tex]
[tex]\[ = -2(x^2 + 2xh + h^2) - x - h - 3 \][/tex]
[tex]\[ = -2x^2 - 4xh - 2h^2 - x - h - 3 \][/tex]
### Step 3: Find the difference [tex]\( f(x+h) - f(x) \)[/tex]
Next, we subtract [tex]\( f(x) \)[/tex] from [tex]\( f(x+h) \)[/tex]:
[tex]\[ f(x+h) - f(x) = (-2x^2 - 4xh - 2h^2 - x - h - 3) - (-2x^2 - x - 3) \][/tex]
Simplify the expression:
[tex]\[ f(x+h) - f(x) = -2x^2 - 4xh - 2h^2 - x - h - 3 + 2x^2 + x + 3 \][/tex]
[tex]\[ = -4xh - 2h^2 - h \][/tex]
### Step 4: Divide by [tex]\( h \)[/tex]
We now divide the entire difference by [tex]\( h \)[/tex]:
[tex]\[ \frac{f(x+h) - f(x)}{h} = \frac{-4xh - 2h^2 - h}{h} \][/tex]
### Step 5: Simplify the expression
Finally, we simplify the expression:
[tex]\[ \frac{f(x+h) - f(x)}{h} = \frac{-4xh}{h} + \frac{-2h^2}{h} + \frac{-h}{h} \][/tex]
[tex]\[ = -4x - 2h - 1 \][/tex]
So, the final simplified expression is:
[tex]\[ \frac{f(x+h)-f(x)}{h} = -4x - 2h - 1 \][/tex]
