Given f(x) = 2x² + 5x-3, find
f(x+h)-f(x)/h.  h≠ 0.

To find \(\frac{f(x+h) - f(x)}{h}\), given \(f(x) = 2x^2 + 5x - 3\), follow these steps:

### Step 1: Find \(f(x + h)\)
We substitute \(x + h\) into the function \(f(x)\):

\[
f(x + h) = 2(x + h)^2 + 5(x + h) - 3
\]

Expand the terms:

\[
= 2(x^2 + 2xh + h^2) + 5(x + h) - 3
\]
\[
= 2x^2 + 4xh + 2h^2 + 5x + 5h - 3
\]

### Step 2: Calculate \(f(x+h) - f(x)\)
Now subtract \(f(x)\) from \(f(x + h)\):

\[
f(x + h) - f(x) = \left(2x^2 + 4xh + 2h^2 + 5x + 5h - 3\right) - \left(2x^2 + 5x - 3\right)
\]

Simplifying this:

\[
= 2x^2 + 4xh + 2h^2 + 5x + 5h - 3 - 2x^2 - 5x + 3
\]
\[
= 4xh + 2h^2 + 5h
\]

### Step 3: Divide by \(h\)
Now, divide the expression by \(h\):

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

Simplifying:

\[
= 4x + 2h + 5
\]

### Final Answer:
\[
\frac{f(x+h) - f(x)}{h} = 4x + 2h + 5
\]

Let me know if you need further clarification or details.

Here are 5 follow-up questions related to this:

1. What happens to the expression as \(h \to 0\)?
2. How would you compute the derivative of \(f(x)\) using the limit definition?
3. How does this result relate to the concept of the difference quotient?
4. Can you find \(f'(x)\) using another method like power rule?
5. How would the expression change if \(f(x)\) were a cubic function?

**Tip:** The expression \(\frac{f(x+h) - f(x)}{h}\) is often called the difference quotient, and it's a foundational concept for finding derivatives.

Given f(x) = 2x² + 5x - 3, find f(x+h) - f(x) / h. h≠ 0.

Learn how to solve the difference quotient for the quadratic function f(x) = 2x² + 5x - 3. This step-by-step solution covers key algebraic concepts such as expanding polynomials and simplifying expressions. Ideal for students in Grades 10-12.

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