To determine the rationalizing factor of the expression [tex]\(\sqrt[3]{25} + \sqrt[3]{5} + 1\)[/tex], we need to find a number or expression that when multiplied with [tex]\(\sqrt[3]{25} + \sqrt[3]{5} + 1\)[/tex] results in a rational expression.
Let’s denote [tex]\( \sqrt[3]{5} \)[/tex] as [tex]\( x \)[/tex]. Hence, we rewrite
[tex]\[ \sqrt[3]{25} + \sqrt[3]{5} + 1 \][/tex]
in terms of [tex]\( x \)[/tex]:
[tex]\[ \sqrt[3]{25} = \sqrt[3]{5^2} = x^2 \][/tex]
So we can rewrite the original expression as:
[tex]\[ x^2 + x + 1 \][/tex]
Now, we are looking for the rationalizing factor of [tex]\( x^2 + x + 1 \)[/tex].
Notice that one of the parts of the expression can be considered a factor that makes the original expression more manageable if cubed, since a cubic root is involved. We need to consider the nature of complex roots of unity and the property that [tex]\((a + b\omega + c\omega^2)\)[/tex] under cubic roots where [tex]\(\omega\)[/tex] is the complex roots. However, without delving deeper into complex conjugates and its thorough arithmetics, the straightforward approach to simplifying and knowing correct rationalizing factor involves knowing unique qualities of cube roots.
From our prior conclusion and the given options, the rationalizing factor for an expression formed mostly around cube roots fits generally to a similar cube root subtraction involving transitively the conjugate nature of common factors' forms.
Therefore, the correct rationalizing factor for [tex]\(\sqrt[3]{25} + \sqrt[3]{5} + 1\)[/tex] is:
[tex]\[
\boxed{\sqrt[3]{5} - 1}
\][/tex]
