How do you factor the expression \[5{{x}^{2}}+16x+3\]?

Hint: In this problem we have to find the factor for the given expression \[5{{x}^{2}}+16x+3\]_. We can first separate the middle term 16 into two terms whose addition is equal to the middle term itself and multiplication is equal to the multiplication of first and the last term. i.e. $5\times 3=15\times 1=15$, which is the multiplication of the first term and the last term. We can then take common terms outside to get the factors.

Complete step by step solution:
We know that the given expression is,
\[5{{x}^{2}}+16x+3\]
We can first split the middle term to form factors.
We have to expand the middle term i.e. the x term with its coefficient in such a way that their addition is equal to the middle term i.e. 16x, and multiplication is equal to \[5\times 3=15\times 1=15\].
\[\Rightarrow 5{{x}^{2}}+15x+1x+3\]
We can now take the first two terms and the last two terms to take common terms outside, we get
\[\Rightarrow \left( 5{{x}^{2}}+15x \right)+\left( 1x+3 \right)\]
Now we can take the common terms outside, we get
\[\Rightarrow 5x\left( x+3 \right)+1\left( x+3 \right)\]
We can again take the common factor first then the remaining terms to make a factor, we get
\[\Rightarrow \left( 5x+1 \right)\left( x+3 \right)\]
Therefore, the factors are \[\left( 5x+1 \right)\left( x+3 \right)\].

Note: Students make mistakes while splitting the middle term into two terms whose addition should be equal to the middle term itself and the multiplication should be equal to the multiplication of first and the last term. We will also make mistakes while taking common factors in which we should concentrate.