How do you find \[2i\left( 1-4i \right)\left( 1+i \right)\]?

Hint: For the question we are asked to find the solution of the \[2i\left( 1-4i \right)\left( 1+i \right)\]. For the questions of this kind we will use the complex number formula which is \[{{i}^{2}}=-1\]. After using the complex number formula we will solve the question using arithmetical operations such as addition and multiplication and solve the question.

Complete step by step solution:
Firstly, for this question we will take the first stage of problem as solving the braces that is \[\left( 1-4i \right)\left( 1+i \right)\] and later after solving these braces in the next stage we multiply it with \[2i\].
So, from the question we have, \[2i\left( 1-4i \right)\left( 1+i \right)\]. In the first stage as we mentioned above the solution will be as follows.
\[\Rightarrow \left( 1-4i \right)\left( 1+i \right)\]
\[\Rightarrow 1\left( 1+i \right)-4i\left( 1+i \right)\]
\[\Rightarrow 1+i-4i-4{{i}^{2}}\]
Here we will use the formula in complex numbers that is \[{{i}^{2}}=-1\] in the above expression and simplify the expression.
After substituting the above formula the equation will be reduced as follows.
\[\Rightarrow 1+i-4i-4\left( -1 \right)\]
Here we know that negative of negative will give us positive. After doing the simplification the expression will be reduced as follows.
\[\Rightarrow 1+i-4i+4\]
\[\Rightarrow \left( 5-3i \right)\]
So, the second stage of simplification will be to multiply \[\Rightarrow \left( 5-3i \right)\] with \[2i\]. After doing the above simplification we will get the solution for the required question \[2i\left( 1-4i \right)\left( 1+i \right)\].
Now, we will multiply \[\left( 5-3i \right)\] with \[2i\]. So, the solution will be as follows.
\[\Rightarrow 2i\left( 5-3i \right)\]
\[\Rightarrow 10i-6{{i}^{2}}\]
\[\Rightarrow 10i-6\left( -1 \right)\]
\[\Rightarrow 10i+6\]
Therefore, the solution for the question will be \[ 10i+6\].

Note: Students must be very careful while solving the questions of this kind. Students must not make any calculation mistakes. Students must have good knowledge in complex numbers concepts. We must do mistake in using the formula of complex number for example if we use \[{{i}^{2}}=1\] instead of \[{{i}^{2}}=-1\] the whole solution will become wrong.