Simplify the expression: 2(x)2 (1/4)xy 8x / ((3x * 4y)^-1)^-2

Step 1: The given expression is
$\frac{2(z^{2} + x^{2} + 4x - 4y - 18z)}{(2x - 4y - 18z)^{2}}.$

Step 2: Factor out the 2 from the base of the denominator:
$2x - 4y - 18z = 2(x - 2y - 9z).$

Step 3: Square the factored form:
$(2x - 4y - 18z)^{2} = [2(x - 2y - 9z)]^{2} = 4(x - 2y - 9z)^{2}.$

Step 4: Substitute into the original expression:
$\frac{2(z^{2} + x^{2} + 4x - 4y - 18z)}{4(x - 2y - 9z)^{2}}.$

Step 5: Simplify the constant coefficient:
$\frac{2}{4} (z^{2} + x^{2} + 4x - 4y - 18z) \cdot \frac{1}{(x - 2y - 9z)^{2}} = \frac{1}{2} \cdot \frac{z^{2} + x^{2} + 4x - 4y - 18z}{(x - 2y - 9z)^{2}}.$

Step 6: Write in single fraction form:
$\frac{z^{2} + x^{2} + 4x - 4y - 18z}{2(x - 2y - 9z)^{2}}.$

\dfrac{z^{2+x^{2}+4x-4y-18z}{2(x-2y-9z)^{2}}}