To eliminate the parameter $\theta$ from the parametric equations $x = 3\cos(\theta) + 1$ and $y = 4\sin(\theta) - 2$, we want to find a relationship between $x$ and $y$ that doesn't involve $\theta$.

Here's the step-by-step process:

1. Express $\cos(\theta)$ and $\sin(\theta)$ in terms of $x$ and $y$:

   • From the first equation, solve for $\cos(\theta)$: $x = 3\cos(\theta) + 1 \implies \cos(\theta) = \frac{x - 1}{3}$
   • From the second equation, solve for $\sin(\theta)$: $y = 4\sin(\theta) - 2 \implies \sin(\theta) = \frac{y + 2}{4}$

2. Use the Pythagorean identity:

   • Recall the identity: $\cos^2(\theta) + \sin^2(\theta) = 1$.
   • Substitute the expressions found for $\cos(\theta)$ and $\sin(\theta)$: $\left(\frac{x - 1}{3}\right)^2 + \left(\frac{y + 2}{4}\right)^2 = 1$

3. Simplify the equation:

   • Expand and simplify: $\frac{(x - 1)^2}{9} + \frac{(y + 2)^2}{16} = 1$

This is the equation of an ellipse with its center at $(1, -2)$, a horizontal semi-axis length of 3, and a vertical semi-axis length of 4.

Summary: The Cartesian equation of the curve is: $\frac{(x - 1)^2}{9} + \frac{(y + 2)^2}{16} = 1$

Would you like a more detailed explanation or have any questions?

Here are some related questions you might find interesting:

1. What are the general steps for eliminating parameters from parametric equations?
2. How can we determine the type of conic section given its Cartesian equation?
3. How do you find the center and axes lengths of an ellipse from its equation?
4. How would the equations change if $\theta$ was eliminated using different trigonometric identities?
5. Can you eliminate the parameter $\theta$ for parametric equations of a hyperbola or parabola?

Tip: When eliminating parameters, always look for trigonometric identities or algebraic manipulation to link the variables directly without the parameter.