Simplify $\dfrac{{{x^3} + 2{x^2} + 3x}}{{2x}}$

Hint: Here we are asked to solve the given algebraic expression in a division format. To make the expression to a simpler format we have to get rid of the term in the denominator that can be done by using the exponent rule that is $\dfrac{{{a^p}}}{{{a^q}}} = {a^{p - q}}$ that is if we have a same base with different powers in the division form then it can be written as the base raised to the power of denominator subtracted from the power of numerator. Then it can be easily simplified to get the required result.

Complete step by step solution:
Given that $\dfrac{{{x^3} + 2{x^2} + 3x}}{{2x}}$ and we need to solve and find the simplified form of the given equation. Since in all the expressions the common variable is $x$ and hence take the variable out of the bracket using the multiplication operation and hence we get $\dfrac{{{x^3} + 2{x^2} + 3x}}{{2x}} = \dfrac{{x({x^2} + 2x + 3)}}{{2x}}$
This is also called the power rule, which is ${x^1} \times {x^2} = {x^{1 + 2}} = {x^3}$ the same base variable, and the power will get added accordingly.
Now using the division operation, cancel the common terms hence we have $\dfrac{{x({x^2} + 2x + 3)}}{{2x}} = \dfrac{{({x^2} + 2x + 3)}}{2}$
Finally, equally given the denominator values to all the numerators, we have \[\dfrac{{({x^2} + 2x + 3)}}{2} = \dfrac{{{x^2}}}{2} + \dfrac{{2x}}{2} + \dfrac{3}{2}\] and cancel out the common terms we get \[\dfrac{{({x^3} + 2{x^2} + 3x)}}{{2x}} = \dfrac{{{x^2}}}{2} + x + \dfrac{3}{2}\] and hence which is the simplified form of the given.

Note: In the above problem, we have used the exponent rule to make the given expression into a simpler form that can also be done by dividing each term in the numerator by the denominator.
Let us consider the given question $\dfrac{{{x^3} + 2{x^2} + 3x}}{{2x}}$ now let us separate this whole expression into three terms by dividing each term in the numerator by the denominator.
$ \Rightarrow \dfrac{{{x^3}}}{{2x}} + \dfrac{{2{x^2}}}{{2x}} + \dfrac{{3x}}{{2x}}$
On simplifying this we get
$ \Rightarrow \dfrac{{{x^2}}}{2} + x + \dfrac{3}{2}$
Hence we got the answer.