Rewrite the rational expression as an equivalent rational expression with the given denominator. (Both are fractions)

-21 = 1
x2 -6x x(x-6)(x+5)

To rewrite the rational expression `-21 / (x^2 - 6x)` as an equivalent rational expression with the denominator `x(x-6)(x+5)`, we need to factor the denominator of the original expression.

First, let's factor the denominator `x^2 - 6x`:
x^2 - 6x = x(x - 6)

Now, we have the denominator factored as `x(x - 6)`. However, we have an additional factor `(x + 5)` in the desired denominator `x(x - 6)(x + 5)`. To obtain this factor, we can multiply both the numerator and denominator of the original expression by `(x + 5)`:

-21 / (x^2 - 6x) = (-21 * (x + 5)) / ((x^2 - 6x) * (x + 5))
= (-21(x + 5)) / (x(x - 6)(x + 5))

Now, the rational expression is equivalent to `-21(x + 5) / (x(x - 6)(x + 5))`. The factor `(x + 5)` in both the numerator and denominator cancels out, leaving us with:

-21(x + 5) / (x(x - 6)(x + 5)) = -21 / (x(x - 6))

To rewrite the rational expression -21 as an equivalent rational expression with the denominator x(x-6)(x+5), we need to find the numerator that corresponds to this denominator.

First, let's factor the denominator x(x-6)(x+5):

x(x-6)(x+5) = x(x^2-x-30)

To find the numerator, we need to multiply -21 by the missing factors in the denominator. The missing factors are x-6 and x+5:

Numerator = -21 * (x-6) * (x+5)

Now we simplify the numerator:

Numerator = -21 * (x^2 - x - 30)

Numerator = -21x^2 + 21x + 630

Therefore, the rational expression -21 can be rewritten as an equivalent rational expression with the denominator x(x-6)(x+5) as:

(-21x^2 + 21x + 630) / (x(x-6)(x+5))