Question

Multiply. State any restrictions on the variable. Show your work.

4x^2+27x+18/16x^2-9 * 4x^2+9x-9/x^2+3x-18

Answer 1

To multiply the given expressions:

(4x^2 + 27x + 18) / (16x^2 - 9) * (4x^2 + 9x - 9) / (x^2 + 3x - 18)

First, factorize the numerator and denominator of each expression:

(4x^2 + 27x + 18) = (4x + 3)(x + 6)
(16x^2 - 9) = (4x + 3)(4x - 3)
(4x^2 + 9x - 9) = (4x - 3)(x + 3)
(x^2 + 3x - 18) = (x + 6)(x - 3)

Now we can cancel out common factors from the numerator and denominator:

((4x + 3)(x + 6) / (4x + 3)(4x - 3)) * ((4x - 3)(x + 3) / (x + 6)(x - 3))

Next, we can simplify further by canceling out the common factors:

(1 / (4x - 3)) * (1 / (x - 3)) = 1 / ((4x - 3)(x - 3))

Therefore, the simplified expression is 1 / ((4x - 3)(x - 3)).

Restrictions on the variable:
The variable x should not equal either 3 or 3/4, as those values would make the denominator zero, resulting in undefined values.