How do you simplify $ \left( { - 1 - 8i} \right) + \left( {4 + 3i} \right) $ and write the complex number in standard form of complex number $ a + ib $ ?

Hint: In the given problem, we are required to simplify an expression involving complex numbers. For simplifying the given expression, we need to have a thorough knowledge of complex number sets and its applications in such questions. Algebraic rules and properties also play a significant role in simplification of such expressions.

Complete step-by-step answer:
In the question, we are given an expression which needs to be simplified using the knowledge of complex number sets. We can also make use of various algebraic properties and rules so as to simplify the expression.
So, we have $ \left( { - 1 - 8i} \right) + \left( {4 + 3i} \right) $
The given expression involves addition of two complex numbers and we have to simplify the sum of these two terms. We should follow BODMAS rules in such questions and give priority to operations accordingly. Now, in order to add up two complex numbers, we need to add their real parts and imaginary parts separately. So, we get,
 $ \Rightarrow \left( { - 1 + 4} \right) + \left( {3i - 8i} \right) $
Opening up the brackets and computing the sum of terms, we get,
 $ \Rightarrow 3 - 5i $
Therefore, the given expression $ \left( { - 1 - 8i} \right) + \left( {4 + 3i} \right) $ can be simplified as: $ 3 - 5i $ and can be written in standard form of complex number as $ a + ib $ with $ a = 3 $ and $ b = - 5 $ .
So, the correct answer is “$ 3 - 5i $”.

Note: The given problem revolves around the application of properties of complex numbers in questions. The question tells us about the wide ranging significance of the complex number set and its properties. The final answer can also be verified by working the solution backwards and getting back the given expression $ \left( { - 1 - 8i} \right) + \left( {4 + 3i} \right) $ . Algebraic rules and properties also play a significant role in simplification of such expressions and we also need to have a thorough knowledge of complex number set and its applications in such questions.