1. **Problem Statement:** The population of mosquitoes in a certain environment follows the differential equation: $$\frac{dx}{dt} = Px^2 + 5$$ where $P$ is a parameter and $t$ ranges from $-4$ to $+4$. 2. **Understanding the equation:** This is a first-order nonlinear differential equation describing the rate of change of the mosquito population $x$ with respect to time $t$. 3. **Given values:** We need to analyze the slope fields for $P = -4$ and $P = +4$ over the domain $t = -4$ to $t = +4$ at whole number grid points. 4. **Slope field calculation:** At each grid point $(t, x)$ where $t$ and $x$ are integers from $-4$ to $4$, compute the slope: $$m = Px^2 + 5$$ For example, when $P = -4$ and $x = 0$, $$m = -4 \times 0^2 + 5 = 5$$ When $P = +4$ and $x = 1$, $$m = 4 \times 1^2 + 5 = 9$$ 5. **Interpretation of slopes:** - For $P = -4$, the term $-4x^2$ decreases the slope as $|x|$ increases, but the constant $+5$ keeps slopes positive for small $x$. - For $P = +4$, the slope increases rapidly with $x^2$, leading to steeper slopes as $|x|$ grows. 6. **Sketching slope elements:** At each grid point, draw a small line segment with slope $m$ calculated above. 7. **Isolines (solution curves):** By connecting these slope elements smoothly, isolines show the behavior of the population over time. 8. **Behavior description:** - For $P = -4$, the population growth rate decreases with larger population sizes due to the negative quadratic term, but the constant term $+5$ ensures some growth. - For $P = +4$, the population growth accelerates rapidly as population increases, indicating possible explosive growth. **Final answer:** The slope fields and isolines reveal that for $P = -4$, the mosquito population grows but is moderated by the negative quadratic term, while for $P = +4$, the population growth accelerates rapidly, potentially leading to uncontrolled increase over the time interval $t = -4$ to $+4$.