Simplify the following rational expressions by factoring and canceling. And define the constraints (what 𝓧 cannot be) (show all work):

Question

38x-48x2-14x+48x-8

Answer 1

To simplify the given rational expression \( 38x - 48x^2 - 14x + 48x - 8 \), we start by combining like terms.

Step 1: Combine Like Terms

We can rearrange and combine the terms for simplification:

\[ 38x - 14x + 48x - 48x^2 - 8 \]

Combining the \(x\) terms:

\[ (38x - 14x + 48x - 48x^2 - 8) = (38x - 14x + 48x) - 48x^2 - 8 \] \[ = (38 - 14 + 48)x - 48x^2 - 8 \] \[ = 72x - 48x^2 - 8 \]

Step 2: Reorganize for Factoring

Reorganizing yields:

\[ -48x^2 + 72x - 8 \]

Step 3: Factor the Expression

First, we notice there’s a common factor in all the coefficients, which is \( -8 \):

\[ -8(6x - 9x + 1) \]

Now we will need to factor further.

\[ -8(-48x^2 + 72x - 8) = -8 \times (6x^2 - 9x + 1) \]

Using the quadratic formula or other factoring methods to factor \(6x^2 - 9x + 1\) might be a good idea. This quadratic does not factor neatly due to its discriminant: \[ D = b^2 - 4ac = (-9)^2 - 4(6)(1) = 81 - 24 = 57 \]

Completing the Simplification:

Since \(6x^2 - 9x + 1\) cannot be factored easily over the integers, we can leave our expression as is or use the quadratic formula for roots if necessary.

Collectively:

\[ = -8(6x^2 - 9x + 1) \]

Step 4: Identify Constraints

We need to ascertain under what conditions this expression is defined (i.e., the denominator cannot equal zero).

In this case, we must ensure the terms at the base of the rational expression don’t lead to division by zero, particularly if any of the factors equate. Since we haven't divided by any expression which could lead to zero in this simplification, there are no constraints imposed by this factorization to address.

Conclusion

Thus, the simplified expression is: \[ -8(6x^2 - 9x + 1) \]

Final Constraints (Where \(x\) cannot be): There are no constraints if there is not a denominator present here. In the general case if this were a part of a rational expression with a true denominator that leads to undefined behavior, consider checking where such denominators equal zero. For this specific expression, none.

To represent full clarity:

1. The simplified output is \( -8(6x^2 - 9x + 1) \).
2. No values of \(x\) will lead to undefined outcomes.