1. **State the problem:** We are given a quadratic function $$F(x) = x^2 + x$$ and two points $$a$$ and $$b=42$$. We want to find the Average Rate of Change (AROC) of $$F(x)$$ between $$a$$ and $$b$$, which is given by the formula: $$\text{AROC} = \frac{F(b) - F(a)}{b - a}$$ 2. **Explain the formula:** The Average Rate of Change between two points on a function measures the average slope of the function between those points. It is the change in the function values divided by the change in the input values. 3. **Apply the formula:** Substitute $$F(x) = x^2 + x$$, $$b=42$$, and unknown $$a$$: $$F(b) = 42^2 + 42 = 1764 + 42 = 1806$$ $$F(a) = a^2 + a$$ So, $$\text{AROC} = \frac{1806 - (a^2 + a)}{42 - a} = \frac{1806 - a^2 - a}{42 - a}$$ 4. **Interpretation:** This expression gives the average rate of change of the function between $$x=a$$ and $$x=42$$. Without a specific value for $$a$$, this is the simplified formula for AROC. 5. **Summary:** The Average Rate of Change between $$a$$ and $$42$$ for the function $$F(x) = x^2 + x$$ is: $$\boxed{\frac{1806 - a^2 - a}{42 - a}}$$ This formula can be used to compute the AROC once $$a$$ is known.