How do you simplify: $ \left( {\dfrac{{10{x^5}}}{{2{x^3} + 2{x^2}}}} \right) $ ?

Hint: As we can clearly see that the question given requires us to simplify a rational algebraic expression $ \left( {\dfrac{{10{x^5}}}{{2{x^3} + 2{x^2}}}} \right) $ . Whenever, we see a question like this, our approach should be to simplify it using algebraic rules and properties like factorisation. Thus, we need to factor out the common factors in numerator and denominator and then cancel them. Then, we write the rational fraction in the simplest form after cancelling the common factor between numerator and denominator.

Complete step-by-step answer:
In the given problem, we are required to find the quotient when $ 10{x^5} $ is divided by the fraction $ \left( {2{x^3} + 2{x^2}} \right) $ .
 So, we first write it in fraction format as: $ \left( {\dfrac{{10{x^5}}}{{2{x^3} + 2{x^2}}}} \right) $
Now, we factor out the common factors of numerator and denominator. So, we get,
 $ \Rightarrow \left( {\dfrac{{2{x^2} \times 5{x^3}}}{{2{x^2}\left( {x + 1} \right)}}} \right) $
Now, we cancel the common factors of numerator and denominator so as to simplify the rational function in x. So, we get,
 $ \Rightarrow \left( {\dfrac{{5{x^3}}}{{x + 1}}} \right) $
Now, we can see that there is no common factor present between the numerator and the denominator of the fraction. So, this is the simplified form that we are required to find.
Hence, the rational function $ \left( {\dfrac{{10{x^5}}}{{2{x^3} + 2{x^2}}}} \right) $ in x given to us in the question can be simplified as $ \left( {\dfrac{{5{x^3}}}{{x + 1}}} \right) $ .
So, the correct answer is “ $ \left( {\dfrac{{5{x^3}}}{{x + 1}}} \right) $ ”.

Note: Such questions which ask us to simplify a rational function in x require us to have a thorough knowledge of algebraic rules and properties. Hence, we should have a strong grip on topics like factorisation and transposition.