Simplify the expression \[\left( {\dfrac{{ - 24}}{{ - 4}}} \right) \div 2 + 3.{( - 5 + 1)^2}\]

Hint: We use the rule called BODMAS to solve this problem and we will know about this rule. Then we will simplify the given expression to get the result.

Complete step by step answer:
In mathematics, we simplify expressions and numbers according to the BODMAS rule.
The full form of BODMAS is
B – Bracket
O – Orders
D – Division
M – Multiplication
A – Addition
S – Subtraction
Now the question is \[\left( {\dfrac{{ - 24}}{{ - 4}}} \right) \div 2 + 3.{( - 5 + 1)^2}\]
First, we need to simplify the terms in brackets.
\[ \Rightarrow \left( 6 \right) \div 2 + 3.{\left( { - 4} \right)^2}\]
Now, we need to simplify order functions.
\[ \Rightarrow 6 \div 2 + 3 \times 16\]
Now, we should go for division operation.
\[ \Rightarrow 3 + 3 \times 16\]
And now, we have to do multiplication operations.
\[ \Rightarrow 3 + 48\]
Now, we have to do addition operation.
\[ \Rightarrow 51\]

Note:
Remember these sign rules which are \[\left( + \right) \times \left( + \right) = \left( + \right)\]; \[\left( + \right) \times \left( - \right) = \left( - \right)\]; \[\left( - \right) \times \left( - \right) = \left( + \right)\] and \[\left( - \right) \times \left( + \right) = \left( - \right)\]
These rules are also applicable for division operation too.
Orders operation is about the radicals or powers. For example, \[{\left( 9 \right)^{\dfrac{1}{2}}}\] is an ordered operation.