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

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

1. **Substitute the function \(g(x)\) and \(g(a)\):**

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

2. **Compute \(g(x) - g(a)\):**

   \[
   g(x) - g(a) = (x^2 + 2x) - (a^2 + 2a)
   \]
   Simplify this expression:
   \[
   g(x) - g(a) = x^2 + 2x - a^2 - 2a
   \]

3. **Factor the difference \(g(x) - g(a)\):**

   Recognize that \(x^2 - a^2\) is a difference of squares:
   \[
   x^2 - a^2 = (x - a)(x + a)
   \]
   So:
   \[
   g(x) - g(a) = (x - a)(x + a) + 2(x - a)
   \]
   Factor out \((x - a)\):
   \[
   g(x) - g(a) = (x - a)(x + a + 2)
   \]

4. **Divide by \(x - a\) (since \(x \neq a\)):**

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

So, the expression simplifies to:

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

Would you like more details on any step or have any questions?

### Related Questions:
1. What is the significance of the expression \(\frac{g(x) - g(a)}{x - a}\) in calculus?
2. How can this expression be interpreted as the slope of the secant line between two points?
3. How would you find the derivative of \(g(x) = x^2 + 2x\) using the definition of a derivative?
4. What happens if \(x = a\) in the expression \(\frac{g(x) - g(a)}{x - a}\)?
5. How can this method be applied to other polynomial functions?
6. What does the expression look like if \(g(x) = x^3 + 3x^2\)?
7. What is the geometric interpretation of this simplification?
8. How does this problem relate to the concept of the difference quotient?

### Tip:
When simplifying expressions involving \(g(x) - g(a)\), look for common patterns like the difference of squares or other factorization techniques to make the process easier.

Learn how to evaluate the expression (g(x) - g(a)) / (x - a) for the quadratic function g(x) = x^2 + 2x. Follow step-by-step solutions and understand the use of the difference quotient formula and the difference of squares theorem. Suitable for advanced high school students.

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