How do you factor $3{{x}^{2}}+36x+81$?

Hint: Now to solve the given problem we will use splitting the middle term method. Hence we will split 36x as 27x + 9x and then we will rewrite the expression. Now we will take 3x common from the first two terms and 9 common from the last two terms. Now on simplifying the expression we will get the factorization of the given expression.

Complete step by step solution:
Now to solve the given expression we will split the middle term.
Now to factorize the expression by this method we will first split the middle terms such that the multiplication of the two terms obtained in the expression is equal to the product of first term and last term.
Now to do so we will split 36x as 27x + 9x.
Now we have $\left( 27x \right)\left( 9x \right)=81\times \left( 3{{x}^{2}} \right)$
Hence we get the given expression as,
$\Rightarrow 3{{x}^{2}}+27x+9x+81$
Now taking 3x common from the first two terms and 9 common from the last two terms we get the expression as,
$\Rightarrow 3x\left( x+9 \right)+9\left( x+9 \right)$
Now again taking x + 9 common from the whole expression we get,
$\Rightarrow \left( x+9 \right)\left( 3x+9 \right)$
Now we cannot further factorize the expression as we have linear expressions.
Hence the given expression is factored.

Note: Now note that we can also first find the roots of the quadratic expression. Now to find the roots of the quadratic expression we can either use the formula $\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$ or use completing the square method and solve. Now once we have the roots of the expression we can write the factors of the expression as $x-\alpha $ where $\alpha $ is the root of the given expression.