How do you factor $3{{x}^{2}}+8x+4$ by using the grouping method?

Hint: We are given $3{{x}^{2}}+8x+4$, we are asked to find the factor form of it using grouping method,
We will learn when grouping method is applied,
We learn about trinomials, we will learn how to split the middle term.
We will split the middle term of the given $3{{x}^{2}}+8x+4$ to make the given polynomial into 4 terms.
Once we get 4 terms, we make a group of 2 terms by considering first 2 terms and last 2 terms at once we take common factors out of these and simplify to get our desired answer.

Complete step by step answer:
We are given $3{{x}^{2}}+8x+4$, we are asked to factor by grouping method.
We will see that our given term is a trinomial.
Trinomial are those which is consist of 3 terms
As we have $3{{x}^{2}}+8x+4$ of 3 terms so it is a trinomial.
Basically a grouping method is applied for 4 terms polynomial. So, when we wish to apply a grouping method over polynomials then we split the middle term into two terms so that we get 4 terms in total, then we re-write and make a pair of the first two terms and the last two. Once we have pairs, we take common factors possible out of each pair.
For an easy calculator we start by taking the greatest common factor out.
Now we are given $3{{x}^{2}}+8x+4$
We can see that all the terms have nothing in common so nothing can be separated out.
So we move to the next step.
We have to split the middle term. To find the perfect split, we multiply the coefficients of ${{x}^{4}}$’a’ and the constant term ‘c’.
Then we find such a pair when give product as $a\times c$ and addition or subtraction of number will form ‘b’.
Now we have $3{{x}^{2}}+8x+4$
So $a=3,c=4$
So, $a\times c=3\times 4=12$
Now $6\times 2=12$ and $6+2=8$
So we use $8=6+2$
So, we get –
$3{{x}^{2}}+8x+4=3{{x}^{2}}+\left( 6+2 \right)x+4$
By opening bracket, we get –
$=3{{x}^{2}}+6x+2x+4$
Now we separate common term from first pair and last pair, we get –
$=3x\left( x+2 \right)+2\left( x+2 \right)$
As $\left( x+2 \right)$ is same so simplifying, we get –
$=\left( 3x+2 \right)\left( x+2 \right)$

So we get factor of $3{{x}^{2}}+8x+4$as $3{{x}^{2}}+8x+4=\left( 3x+2 \right)\left( x+2 \right)$.

Note: Key points to remember are that the sign of coefficient of leading entry and the constant are the same, then we find the split of ‘b’ by adding the two terms. But if we have signs of these entries opposite to each other then we split the term ‘b’ by subtracting two terms. If we do not follow this then we end up on the wrong path.