To solve $\frac{f(x+h) - f(x)}{h}$ for the function $f(x) = 2x^2 + 5x - 3$, let's go step by step.

Step 1: Find $f(x+h)$

We need to substitute $x + h$ into the function $f(x)$.

$f(x+h) = 2(x+h)^2 + 5(x+h) - 3$

Now expand the terms: $= 2(x^2 + 2xh + h^2) + 5(x + h) - 3$ $= 2x^2 + 4xh + 2h^2 + 5x + 5h - 3$

Step 2: Subtract $f(x)$

Now subtract $f(x) = 2x^2 + 5x - 3$ from $f(x+h)$:

$f(x+h) - f(x) = \left( 2x^2 + 4xh + 2h^2 + 5x + 5h - 3 \right) - \left( 2x^2 + 5x - 3 \right)$ Simplify by canceling out the common terms: $f(x+h) - f(x) = 4xh + 2h^2 + 5h$

Step 3: Divide by $h$

Now, divide the result by $h$:

$\frac{f(x+h) - f(x)}{h} = \frac{4xh + 2h^2 + 5h}{h}$

Factor out $h$ in the numerator: $= \frac{h(4x + 2h + 5)}{h}$

Since $h \neq 0$, we can cancel out $h$:

$= 4x + 2h + 5$

Final Answer:

$\frac{f(x+h) - f(x)}{h} = 4x + 2h + 5$

Would you like a more detailed explanation or have any questions?

Here are some related questions to practice:

1. What is the limit of $\frac{f(x+h) - f(x)}{h}$ as $h \to 0$?
2. Can you find the derivative of $f(x)$?
3. What is the equation of the tangent line to the graph of $f(x)$ at $x = 2$?
4. How would you compute $\frac{f(x+2) - f(x)}{2}$?
5. What does $f'(x)$ tell you about the graph of $f(x)$?

Tip: When simplifying, always look for common terms to cancel out—it makes calculations much easier!