The image shows the following problem:

You are asked to evaluate the difference quotient for the function $f(x) = x^2 - 2x + 3$. The difference quotient is given by:

$\frac{f(a + h) - f(a)}{h}$

Let's solve this step by step:

1. Find $f(a + h)$: $f(a + h) = (a + h)^2 - 2(a + h) + 3$ Expanding this: $(a + h)^2 = a^2 + 2ah + h^2$ $-2(a + h) = -2a - 2h$ Thus, $f(a + h) = a^2 + 2ah + h^2 - 2a - 2h + 3$

2. Find $f(a)$: $f(a) = a^2 - 2a + 3$

3. Compute $f(a + h) - f(a)$: $f(a + h) - f(a) = \left(a^2 + 2ah + h^2 - 2a - 2h + 3\right) - \left(a^2 - 2a + 3\right)$ Simplifying this expression: $f(a + h) - f(a) = 2ah + h^2 - 2h$

4. Divide by $h$: $\frac{f(a + h) - f(a)}{h} = \frac{2ah + h^2 - 2h}{h}$ Factor out $h$ in the numerator: $\frac{f(a + h) - f(a)}{h} = \frac{h(2a + h - 2)}{h}$ Cancel $h$ (as long as $h \neq 0$): $\frac{f(a + h) - f(a)}{h} = 2a + h - 2$

So, the difference quotient for the function $f(x) = x^2 - 2x + 3$ is:

$2a + h - 2$

Would you like further details or clarifications? Here are 5 related questions that can expand your understanding:

1. How would you evaluate the limit of the difference quotient as $h \to 0$?
2. What does the difference quotient represent in calculus?
3. How would the result change if the function were cubic instead of quadratic?
4. Can you compute the derivative of the function using this result?
5. How does the difference quotient relate to the slope of a secant line?

Tip: The difference quotient is the foundation of the derivative. As $h \to 0$, the difference quotient gives the slope of the tangent line to the curve at $x = a$.