1. **State the problem:** We need to find and simplify the expression $$\frac{f(a+h)-f(a)}{h}$$ where $$f(x) = 4x + 7$$. 2. **Recall the formula:** This expression represents the difference quotient, which is used to find the average rate of change of the function over the interval from $$a$$ to $$a+h$$. 3. **Calculate $$f(a+h)$$:** Substitute $$x = a+h$$ into $$f(x)$$: $$f(a+h) = 4(a+h) + 7 = 4a + 4h + 7$$ 4. **Calculate $$f(a)$$:** Substitute $$x = a$$ into $$f(x)$$: $$f(a) = 4a + 7$$ 5. **Form the difference quotient:** $$\frac{f(a+h)-f(a)}{h} = \frac{(4a + 4h + 7) - (4a + 7)}{h}$$ 6. **Simplify the numerator:** $$4a + 4h + 7 - 4a - 7 = 4h$$ 7. **Rewrite the expression:** $$\frac{4h}{h}$$ 8. **Cancel the common factor $$h$$:** $$\frac{\cancel{4h}}{\cancel{h}} = 4$$ 9. **Final answer:** $$\boxed{4}$$ This means the average rate of change of the function $$f(x) = 4x + 7$$ over any interval is 4, which is also the slope of the line.