How do you factor $3{{x}^{2}}+16x+5$?

Hint: To find the factors of $3{{x}^{2}}+16x+5$, first multiply the coefficient of \[{{x}^{2}}\] i.e. ‘3’ with ‘5’. Then find two numbers whose sum or difference will be the coefficient of ‘x’ i.e. ‘16’ and multiplication will be the result of multiplication of ‘3’ and ‘5’ i.e. ‘15’. Then break the equation and take common factors out and group them together to obtain the required solution.

Complete step by step solution:
Factorization: Factorization is defined as the breaking or decomposition of an expression into a product of other factors, which when multiplied together gives the original expression itself.
The equation we have, $3{{x}^{2}}+16x+5$
Multiplying the coefficient of \[{{x}^{2}}\] with the constant term, we get $3\times 5=15$
Now, we have to find two numbers whose sum is ‘16’ and multiplication is ‘16’.
Thus, the numbers are ‘1’ and ‘15’.
Hence, our equation can be written as
$\begin{align}
  & 3{{x}^{2}}+16x+5 \\
 & \Rightarrow 3{{x}^{2}}+1x+15x+5 \\
\end{align}$
Taking common ‘x’ from first two terms and ‘5’ from last two terms, we get
$\Rightarrow x\left( 3x+1 \right)+5\left( 3x+1 \right)$
Again taking common $\left( 3x+1 \right)$ from both the terms, we get
$\Rightarrow \left( 3x+1 \right)\left( x+5 \right)$
This is the required solution of the given question.

Note: In the factorization method, we can reduce any algebraic or quadratic equation into its simpler form, where the equations are represented as the product of factors. The given expression can also be factorized by reducing the coefficient of ‘\[{{x}^{2}}\]’ (by taking the coefficient of \[{{x}^{2}}\] common out from all the terms) and then applying completing square method.