f(x+h)-f(x)÷h=x²+2xh+h²+x+h+3-(x²+x+3)

To solve the given expression:

\[
\frac{f(x+h) - f(x)}{h} = x^2 + 2xh + h^2 + x + h + 3 - (x^2 + x + 3)
\]

First, simplify the numerator:

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

Now, distribute and combine like terms:

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

The \( x^2 \) and \( -x^2 \) terms cancel out, as do the \( x \) and \( -x \) terms, and the \( 3 \) and \( -3 \) terms:

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

Next, divide the entire expression by \( h \):

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

Separate the terms in the numerator:

\[
\frac{2xh}{h} + \frac{h^2}{h} + \frac{h}{h}
\]

Simplify each term:

\[
= 2x + h + 1
\]

Therefore, we have:

\[
\frac{f(x+h) - f(x)}{h} = 2x + h + 1
\]

This simplification shows the expression in its final form.

Do you want more details or have any questions?

Here are some related questions you might find interesting:
1. How does the limit of \(\frac{f(x+h) - f(x)}{h}\) as \(h\) approaches 0 relate to the derivative of \(f(x)\)?
2. What is the derivative of \(f(x) = x^2 + x + 3\)?
3. How do you find the derivative of a polynomial function?
4. What role does the term \(h^2\) play in the limit definition of a derivative?
5. How can you interpret the term \(2xh\) geometrically?
6. Why do the constant terms cancel out when simplifying \(f(x+h) - f(x)\)?
7. How does the expression change if \(f(x)\) includes higher degree terms like \(x^3\)?
8. Can you generalize the result for any quadratic polynomial function?

**Tip:** When finding the limit as \(h\) approaches 0 for \(\frac{f(x+h) - f(x)}{h}\), always simplify the numerator first to make it easier to divide by \(h\).

Explore how to solve a mathematical problem involving the difference quotient and limits in calculus. This detailed solution covers the application of limits and derivatives to polynomials, emphasizing key concepts like functions, limits, and derivatives. Suitable for students in Grades 11-12.

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