Did I do it right? And will i need to find the value of C?
That rationalization is incorrect. It is a cube root not a square root.
You need to use $\displaystyle \left( {\frac{{\sqrt[3]{{{{\left( {1 + cx} \right)}^2}}} + \sqrt[3]{{\left( {1 + cx} \right)}} + 1}}{{\sqrt[3]{{{{\left( {1 + cx} \right)}^2}}} + \sqrt[3]{{\left( {1 + cx} \right)}} + 1}}} \right)$
I have no idea what FOIL means. Is it some term from some dumb math-ed class?
Just learn to do basic algebra.
Surely you know that $\displaystyle a^3-b^3=(a-b)(a^2+ab+b^2)~!$
$\displaystyle \left( {\frac{{\sqrt[3]{{1 + cx}} - 1}}{x}} \right)\left( {\frac{{\sqrt[3]{{{{\left( {1 + cx} \right)}^2}}} + \sqrt[3]{{\left( {1 + cx} \right)}} + 1}}{{\sqrt[3]{{{{\left( {1 + cx} \right)}^2}}} + \sqrt[3]{{\left( {1 + cx} \right)}} + 1}}} \right) = \frac{c}{{\sqrt[3]{{{{\left( {1 + cx} \right)}^2}}} + \sqrt[3]{{\left( {1 + cx} \right)}} + 1}}$