How do you simplify the expression: \[\dfrac{\left( 12{{m}^{4}}-24{{m}^{3}}+6{{m}^{2}}+9m-9 \right)}{6{{m}^{2}}}\]?

Hint: Take the common factor 3 common from all the terms in the numerator and cancel it with the common factors present in the denominator. Now, break all the terms and simplify the expression using the basic formulas of the topic ‘exponents and powers’ given as: - \[\dfrac{{{x}^{a}}}{{{x}^{b}}}={{x}^{a-b}}\] and \[\dfrac{1}{{{x}^{a}}}={{x}^{-a}}\].

Complete step by step solution:
Here, we have been provided with the expression: \[\dfrac{\left( 12{{m}^{4}}-24{{m}^{3}}+6{{m}^{2}}+9m-9 \right)}{6{{m}^{2}}}\] and we have been asked to simplify it. Now, let us assume the given algebraic expression as ‘E’ and use some basic formulas of exponents to simplify it. So, we have,
\[\Rightarrow E=\dfrac{12{{m}^{4}}-24{{m}^{3}}+6{{m}^{2}}+9m-9}{6{{m}^{2}}}\]
Clearly, we can see that the factor 3 is common in all the terms present in the numerator, so taking 3 common we get,
\[\Rightarrow E=\dfrac{3\left( 4{{m}^{4}}-8{{m}^{3}}+2{{m}^{2}}+3m-3 \right)}{6{{m}^{2}}}\]
Cancelling the common factors from the numerator and denominator we get,
\[\Rightarrow E=\dfrac{4{{m}^{4}}-8{{m}^{3}}+2{{m}^{2}}+3m-3}{2{{m}^{2}}}\]
Now, breaking all the terms we have the expression as: -
\[\Rightarrow E=\dfrac{4{{m}^{4}}}{2{{m}^{2}}}-\dfrac{8{{m}^{4}}}{2{{m}^{2}}}+\dfrac{2{{m}^{2}}}{2{{m}^{2}}}+\dfrac{3m}{2{{m}^{2}}}-\dfrac{3}{2{{m}^{2}}}\]
Here, cancelling the common factors and using the formula: - \[\dfrac{{{x}^{a}}}{{{x}^{b}}}={{x}^{a-b}}\] and \[\dfrac{1}{{{x}^{a}}}={{x}^{-a}}\], we get,
\[\begin{align}
  & \Rightarrow E=2{{m}^{4-2}}-4{{m}^{3-2}}+{{m}^{2-2}}+\dfrac{3}{2}{{m}^{1-2}}-\dfrac{3}{2}{{m}^{-2}} \\
 & \Rightarrow E=2{{m}^{2}}-4m+{{m}^{0}}+\dfrac{3}{2}{{m}^{-1}}-\dfrac{3}{2}{{m}^{-2}} \\
\end{align}\]
We know that \[{{m}^{0}}=1\], so we have,
\[\begin{align}
  & \Rightarrow E=2{{m}^{2}}-4m+1+\dfrac{3}{2m}-\dfrac{3}{2{{m}^{2}}} \\
 & \Rightarrow E=2{{m}^{2}}-4m+1+\dfrac{3}{2m}\left( 1-\dfrac{1}{m} \right) \\
\end{align}\]
Hence, the above relation is the simplified form and our answer.

Note:
One may note that here we are unable to factorize the expression present in the numerator because it is a biquadratic equation and we need to determine two roots by the hit – and – trial method so that it can be converted into a quadratic expression whose factored form can be easily determine by the middle term split method. However, the main problem here is to find the two roots using the hit – and – trial method. You must remember some basic formulas of the topic ‘exponents and powers’ and some basic algebraic identities, as they are used everywhere.