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**Hint:**Here it is sufficient to find the range of $a$for the required answer. Consider the three real roots in G.P. and then use \[{\text{Product of roots = }}-\dfrac{{{\text{constant term}}}}{{{\text{coefficient of }}{x^3}}}\] and \[{\text{Sum of roots = }}-\dfrac{{{\text{coefficient of }}{x^2}}}{{{\text{coefficient of }}{x^3}}}\] to find the range of $a$. Then, compare the given options with the range of $a$ to find the correct answer.

**Complete step-by-step answer:**

It is given that these roots are in G.P. Let the common difference of this G.P. be $r$.

Let us consider the roots of the equation be $\dfrac{k}{r}$, $r$ and $kr$ for simplicity.

For a third-degree equation with real roots, it is known that the product of the real roots is equal to the \[-\dfrac{{{\text{constant term}}}}{{{\text{coefficient of }}{x^3}}}\].

Here the constant term is \[ - 1\] , the real roots are $\dfrac{k}{r}$, $r$ and $kr$ and coefficient of \[{x^3}\] is 8.

Therefore

$\dfrac{k}{r} \times r \times kr = - \dfrac{{\left( { - 1} \right)}}{8}$

${k^3} = \dfrac{1}{8}$

$k = \dfrac{1}{2}$

Thus the three roots become

$\dfrac{1}{{2r}}$, $\dfrac{1}{2}$ and $\dfrac{r}{2}$.

Also, it is known that the sum of the real roots of the third-degree polynomial is equal to the \[-\dfrac{{{\text{coefficient of }}{x^2}}}{{{\text{coefficient of }}{x^3}}}\].

Here the coefficient of \[{x^2}\] is \[ - a\], the real roots are $\dfrac{1}{{2r}}$, $\dfrac{1}{2}$,$\dfrac{r}{2}$ and coefficient of \[{x^3}\] is 8.

Therefore ,

\[\dfrac{1}{{2r}} + \dfrac{1}{2} + \dfrac{r}{2} = - \dfrac{{\left( { - a} \right)}}{8}\]

Multiplying the equation throughout with 8 we get

$

\dfrac{4}{r} + 4 + 4r = a \\

a = 4 + 4\left( {\dfrac{1}{r} + r} \right) \\

$

Here the range of a can be defined by defining the scope of the function \[\dfrac{1}{r} + r\].

For \[r > 0\], the function \[\dfrac{1}{r} + r\] ranges from 2 to \[\infty \] with its minimum value, 2 at \[r = 1\].

And the range of \[a\]for \[r > 0\]is therefore defined by

\[a = 4 + 4\left( {\dfrac{1}{r} + r} \right)\]

For \[r > 0\],

Min value of \[a = 4 + 4\left( 2 \right)\]

\[a = 12\]

For \[r > 0\], \[a \in \left( {12,\infty } \right)\]

For \[r < 0\], the function \[\dfrac{1}{r} + r\] ranges from \[ - \infty \] to –2 with its minimum value (–2) occuring at \[r = - 1\].

And the range of \[a\] for \[r < 0\] is therefore defined by

\[a = 4 + 4\left( {\dfrac{1}{r} + r} \right)\]

For \[r < 0\],

Min value of \[a = 4 + 4\left( { - 2} \right)\]

\[a = - 4\]

For \[r < 0\], \[a \in \left( { - \infty , - 4} \right)\].

On combining the range,

\[a \in \left( { - \infty , - 4} \right) \cup \left( {12,\infty } \right)\]

Comparing the range of with the given options, we can see that the option 3 matches the range.

Hence, option C is the correct answer.

**Note:**While taking the roots in G.P., choose numbers such as $\dfrac{k}{r}$, $r$ and $kr$ to avoid tricky calculations. Formulate the equations using the formulas \[{\text{Product of roots = }}-\dfrac{{{\text{constant term}}}}{{{\text{coefficient of }}{x^3}}}\] and \[{\text{Sum of roots = }}-\dfrac{{{\text{coefficient of }}{x^2}}}{{{\text{coefficient of }}{x^3}}}\].

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