Step-by-step Solution:

1. Write the van der Waals Equation: The van der Waals equation for one mole of a gas is given by: $\left(P + \frac{a}{V_m^2}\right)(V_m - b) = RT$ where $V_m$ is the molar volume, $P$ is the pressure, $T$ is the temperature, $R$ is the universal gas constant, and $a$ and $b$ are the van der Waals constants.

2. Rearrange into a Cubic Equation in $V_m$: To find the coefficients of the powers of $V_m$, we need to expand the equation and arrange it as a polynomial in $V_m$.

First, multiply both sides by $V_m^2$ to eliminate the fraction: $\left(P V_m^2 + a\right)(V_m - b) = RT V_m^2$

Next, expand the left side of the equation: $P V_m^3 - P b V_m^2 + a V_m - ab = RT V_m^2$

Now, bring all terms to one side to form a standard cubic equation ($Ax^3 + Bx^2 + Cx + D = 0$): $P V_m^3 - P b V_m^2 - RT V_m^2 + a V_m - ab = 0$

Group the terms with the same power of $V_m$: $(P)V_m^3 - (Pb + RT)V_m^2 + (a)V_m - ab = 0$

3. Identify the Coefficients: From the cubic equation derived above, we can identify the coefficients:

• Coefficient of $V_m^2$: $-(Pb + RT)$
• Coefficient of $V_m$: $a$

4. Determine the Required Ratio: The question asks for the ratio of the coefficient of $V_m^2$ to the coefficient of $V_m$. $\text{Ratio} = \frac{\text{Coefficient of } V_m^2}{\text{Coefficient of } V_m} = \frac{-(Pb + RT)}{a}$

5. Substitute the Given Values: We are given the following values:

• $P = 300 \text{ atm}$
• $T = 300 \text{ K}$
• $a = 6.0 \text{ dm}^6 \text{ atm mol}^{-2}$
• $b = 0.060 \text{ dm}^3 \text{ mol}^{-1}$
• $R = 0.082 \text{ dm}^3 \text{ atm mol}^{-1} \text{ K}^{-1}$

6. Calculate the Ratio: First, let's calculate the term $(Pb + RT)$: $Pb = (300 \text{ atm}) \times (0.060 \text{ dm}^3 \text{ mol}^{-1}) = 18.0 \text{ dm}^3 \text{ atm mol}^{-1}$ $RT = (0.082 \text{ dm}^3 \text{ atm mol}^{-1} \text{ K}^{-1}) \times (300 \text{ K}) = 24.6 \text{ dm}^3 \text{ atm mol}^{-1}$ $Pb + RT = 18.0 + 24.6 = 42.6 \text{ dm}^3 \text{ atm mol}^{-1}$

Now, substitute this value and the value of $a$ into the ratio expression: $\text{Ratio} = \frac{-(42.6 \text{ dm}^3 \text{ atm mol}^{-1})}{6.0 \text{ dm}^6 \text{ atm mol}^{-2}}$ $\text{Ratio} = -7.1 \frac{\text{dm}^3 \cdot \text{atm} \cdot \text{mol}^{-1}}{\text{dm}^6 \cdot \text{atm} \cdot \text{mol}^{-2}} = -7.1 \text{ mol dm}^{-3}$

7. Final Answer: The calculated ratio is -7.1 mol dm$^{-3}$. This value lies within the acceptable range of -7.2 to -7.0.