Verify the following;
\[\left( { - 21} \right) \times \left[ {\left( { - 4} \right) + \left( { - 6} \right)} \right] = \left[ {\left( { - 21} \right) \times \left( { - 4} \right)} \right] + \left[ {\left( { - 21} \right) \times \left( { - 6} \right)} \right]\]

Hint: In this question solve the left hand and the right hand separately and prove them equal. Use the concept that ‘+’ multiplied by ‘-’ is negative. First solve the inside bracket part and then multiply it with the out simplified number.

Complete step-by-step answer:
Given expression is

\[\left( { - 21} \right) \times \left[ {\left( { - 4} \right) + \left( { - 6} \right)} \right] = \left[ {\left( { - 21} \right) \times \left( { - 4} \right)} \right] + \left[ {\left( { - 21} \right) \times \left( { - 6} \right)} \right]\]

Consider L.H.S

\[ \Rightarrow \left( { - 21} \right) \times \left[ {\left( { - 4} \right) + \left( { - 6} \right)}

\right]\]

Now simplify the above equation (according to property that ‘+’ multiplied by ‘-’ is negative)

we have,

\[ \Rightarrow \left( { - 21} \right) \times \left[ { - 4 - 6} \right]\]

$ \Rightarrow \left( { - 21} \right)\left( { - 10} \right)$

Now negative-negative multiplication is positive so we have,

$ \Rightarrow \left( { - 21} \right)\left( { - 10} \right) = 210$

Now consider R.H.S

\[ \Rightarrow \left[ {\left( { - 21} \right) \times \left( { - 4} \right)} \right] + \left[ {\left( { - 21}

\right) \times \left( { - 6} \right)} \right]\]

Now simplify the above equation (according to property that negative-negative
multiplication is positive) we have,

\[ \Rightarrow \left[ {84} \right] + \left[ {126} \right]\]

$ \Rightarrow 84 + 126 = 210$

So as we see that L.H.S = R.H.S

Hence verified.

Note: One concept that is very favored while solving such problem statements is the concept of BODMAS. BODMAS stands for Bracket, Of, Division, Multiplication, Addition and Subtraction. According to this rule, the brackets have to be solved first followed by powers or roots (I.e. Of), then division and then so on according to the name provided.