Find the difference quotient of \( f \), that is, find \( \frac{f(x+h)-f(x)}{h}, h \neq 0 \), for the following function. Be sure to simplify. \[ f(x)=x^{2}-6 x+7 \] \[ \frac{f(x+h)-f(x)}{h}=\square \text { (Simplify your answer.) } \]

To find the difference quotient \( \frac{f(x+h)-f(x)}{h} \), we first calculate \( f(x+h) \): \[ f(x+h) = (x+h)^{2} - 6(x+h) + 7 \] Expanding \( f(x+h) \): \[ = x^{2} + 2xh + h^{2} - 6x - 6h + 7 \] Now, substituting this into the difference quotient: \[ f(x+h) - f(x) = (x^{2} + 2xh + h^{2} - 6x - 6h + 7) - (x^{2} - 6x + 7) \] Simplifying this gives: \[ = 2xh + h^{2} - 6h \] Now we can write the difference quotient: \[ \frac{f(x+h) - f(x)}{h} = \frac{2xh + h^{2} - 6h}{h} \] We can factor out \( h \) from the numerator: \[ = \frac{h(2x + h - 6)}{h} \] Since \( h \neq 0 \), we can cancel \( h \): \[ = 2x + h - 6 \] So the simplified difference quotient is: \[ \frac{f(x+h)-f(x)}{h} = 2x + h - 6 \]