We need to construct a rational expression that has non-permissible values (values that make the denominator zero) of $x = -4, 0,$ and $3$, and simplifies to $\frac{6x+30}{7x-21}$.

1. Problem Description

We need to construct a rational expression that has non-permissible values (values that make the denominator zero) of $x = -4, 0,$ and $3$, and simplifies to $\frac{6x+30}{7x-21}$.

2. Solution Steps

First, let's factor the expression to which we want to simplify:

$$\frac{6x+30}{7x-21} = \frac{6(x+5)}{7(x-3)}$$

Since the non-permissible values are $-4, 0,$ and $3$, we want to construct a denominator that contains the factors $(x+4)$, $(x)$, and $(x-3)$.

Let's consider the following expression:

$$\frac{6(x+5)x(x+4)}{7(x-3)x(x+4)}$$

When simplified, this expression becomes:

$$\frac{6(x+5)}{7(x-3)} = \frac{6x+30}{7x-21}$$

The non-permissible values of the original expression are the values of $x$ for which the denominator is zero:

$$7(x-3)x(x+4) = 0$$

$$x = 3, 0, -4$$

Thus, the expression we constructed meets the criteria.

3. Final Answer

The expression is $\frac{6x(x+4)(x+5)}{7x(x+4)(x-3)}$. This simplifies to $\frac{6x+30}{7x-21}$ and has non-permissible values of -4, 0, and 3.