Given the function  𝑔(𝑥)=𝑥2+2𝑥,
evaluate  𝑔(𝑥)−𝑔(𝑎)𝑥−𝑎,𝑥≠𝑎.

To evaluate the expression \(\frac{g(x) - g(a)}{x - a}\) given the function \(g(x) = x^2 + 2x\), follow these steps:

### Step 1: Compute \(g(x)\)
Given \(g(x) = x^2 + 2x\), the function is already defined.

### Step 2: Compute \(g(a)\)
Substitute \(a\) for \(x\) in the function \(g(x)\):

\[
g(a) = a^2 + 2a
\]

### Step 3: Find the difference \(g(x) - g(a)\)
Subtract \(g(a)\) from \(g(x)\):

\[
g(x) - g(a) = (x^2 + 2x) - (a^2 + 2a)
\]

Distribute and simplify:

\[
g(x) - g(a) = x^2 + 2x - a^2 - 2a
\]

### Step 4: Factor the expression
Notice that the difference of squares can be factored:

\[
g(x) - g(a) = (x^2 - a^2) + 2(x - a)
\]

The difference of squares \(x^2 - a^2\) can be factored as:

\[
g(x) - g(a) = (x - a)(x + a) + 2(x - a)
\]

Now, factor out \((x - a)\) from the entire expression:

\[
g(x) - g(a) = (x - a)(x + a + 2)
\]

### Step 5: Simplify \(\frac{g(x) - g(a)}{x - a}\)
Finally, divide by \((x - a)\) (since \(x \neq a\), we can divide by \((x - a)\)):

\[
\frac{g(x) - g(a)}{x - a} = x + a + 2
\]

### Final Answer
\[
\frac{g(x) - g(a)}{x - a} = x + a + 2
\]

This is the simplified form of the expression.

Would you like more details or have any questions?

### Follow-up Questions
1. How would you evaluate the expression if \(g(x)\) was a different polynomial?
2. What happens if \(g(x)\) is a higher degree polynomial?
3. How is this expression related to the derivative of the function \(g(x)\)?
4. Can you find the derivative of \(g(x)\) using the limit definition?
5. What would be the result if \(a = 0\) in the given expression?
6. How would the expression change if \(g(x) = x^3 + 3x\)?
7. What is the geometric interpretation of this expression?
8. How does this expression relate to the concept of the slope of a secant line?

**Tip:** The expression \(\frac{g(x) - g(a)}{x - a}\) is often used in calculus as a difference quotient, which is closely related to the derivative of the function.

Learn how to evaluate the expression (g(x) - g(a)) / (x - a) using the quadratic function g(x) = x^2 + 2x. This problem covers the difference quotient formula in polynomials and is suitable for high school students.

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