Find the Product:
\[ \left(\frac{2}{x} + 3x\right) \left(\frac{4}{x^2} + 9x^2 – 6\right) \]
Solution:
Rearranging the terms:
\[ \left(\frac{2}{x} + 3x\right) \left(\frac{4}{x^2} – \frac{6}{1} + 9x^2\right) \]
Using identity:
\[ (a+b)(a^2-ab+b^2)=a^3+b^3 \]
Here, \[ a=\frac{2}{x},\qquad b=3x \]
\[ \left(\frac{2}{x} + 3x\right) \left[\left(\frac{2}{x}\right)^2 – \left(\frac{2}{x}\right)(3x) + (3x)^2\right] \]
\[ = \left(\frac{2}{x}\right)^3 + (3x)^3 \]
\[ = \frac{8}{x^3} + 27x^3 \]