what is the quotient in simplified form? State any restrictions on the variable. (6-x)/x^2+3x-28 divided x^2-36/x^2+5x-36

(6-x)/(x^2+3x-28) ÷ ( (x^2-36)/(x^2+5x-36) )

= (6-x)/(x^2+3x-28) * (x^2+5x-36)/(x^2 - 36)
= (6-x)/( (x+7)(x-4) ) * (x+9)(x-4)/((x-6)(x+6) )
= -(x+9)/( (x+7)(x+6) ) , x ≠ 4,6

The restriction of x ≠ 4,6 must be stated, otherwise we actually did divide by zero.
The restriction of x ≠ -7, -6 is implied, in the original problem as well as the answer, it doesn't really have to be stated

{(6-x)/[(x+7)(x-4)]} / {[(x+6)(x-6)]/[(x+9)(x-4)]} = [(6-x)(x+9)(x-4)] / [(x+7)(x-4)(x+6)(x-6)] =

-(x+9) / [(x+7)(x+6)]

x cannot equal -6 or -7 ... division by zero is a no-no

To simplify the expression (6-x)/(x^2+3x-28) ÷ (x^2-36)/(x^2+5x-36), we can follow these steps:

1. Factor the denominators:
x^2+3x-28 = (x-4)(x+7)
x^2-36 = (x-6)(x+6)
x^2+5x-36 = (x-4)(x+9)

2. Rewrite the expression as a multiplication:
(6-x)/(x-4)(x+7) * (x+6)/(x-6)(x+9)

3. Flip the second fraction and change the operation to multiplication:
(6-x)/(x-4)(x+7) * (x+6)/(x-6)(x+9) * (x-6)(x+9)/(x+6)

4. Cancel out the common factors:
(6-x) / (x-4) * (x-6) / (x+7)

5. Simplify further:
(6-x)(x-6) / (x-4)(x+7)

This is the simplified form of the quotient. As for the restrictions on the variable, we need to consider the denominators. The expression is undefined when any of the denominators are equal to zero. Thus, we have two restrictions:
x-4 ≠ 0 (x ≠ 4)
x+7 ≠ 0 (x ≠ -7)

Therefore, the variable x cannot be equal to 4 or -7.

To find the quotient in simplified form, we need to perform division on the given expression: (6 - x) / (x^2 + 3x - 28) divided by (x^2 - 36) / (x^2 + 5x - 36).

First, let's factor both denominators.

The denominator of the dividend (the expression being divided) is x^2 + 3x - 28. This can be factored as (x + 7)(x - 4).

The denominator of the divisor (the expression doing the dividing) is x^2 - 36. This is a difference of squares and can be factored as (x + 6)(x - 6).

Now we can rewrite the division expression as:

[(6 - x) / (x + 7)(x - 4)] ÷ [(x + 6)(x - 6) / (x^2 + 5x - 36)].

Next, we can simplify the division by multiplying the dividend by the reciprocal of the divisor. Reciprocating a fraction means swapping the numerator and the denominator.

So, our expression now becomes:

[(6 - x) / (x + 7)(x - 4)] × [(x^2 + 5x - 36) / (x + 6)(x - 6)].

Now, we can cancel out common factors in the numerator and denominator.

We have (x + 6) and (x - 6) in both the numerator and denominator, so they cancel out.

Our expression simplifies to:

[(6 - x) / (x + 7)(x - 4)] × [(x + 5)(x - 7) / 1].

We can further simplify this to:

(6 - x)(x + 5)(x - 7) / (x + 7)(x - 4).

The quotient in simplified form is: (6 - x)(x + 5)(x - 7) / (x + 7)(x - 4).

As for the restrictions on the variable, we need to consider any values that would make the denominator equal to zero. In this case, both denominators, (x + 7)(x - 4) and (x + 6)(x - 6), should not be equal to zero.

So, the restrictions on the variable are x ≠ -7, x ≠ 4, x ≠ -6, and x ≠ 6.