Expand the expression $\left(a-2b-3c\right)^2$

Hint: In the given question, we need to expand the given expression. By using \[\ \left( x + y + c \right)^{2}\] formula, we can expand the expression. Then by removing the parentheses and multiplying the terms we get the expansion of \[\ \left( a – 2b – 3c \right)^{2}\]
Formula used :
\[{\ \left( x + y + z \right)}^{2} = x^{2} + y^{2} + z^{2} + 2xy + 2yz + 2zx\]

Complete step-by-step solution:
Given, \[\left( a – 2b – 3c \right)^{2}\]
By rearranging the expression according to the formula,
We get,
\[\left( a – 2b – 3c \right)^{2} = \left( a + \left( - 2b \right) + \left( - 3c \right) \right)^{2}\]
Comparing it with the formula,
We get , \[x = \ a\]
\[y = - 2b\]
\[z = - 3c\]
By applying this values in the formula,
\[\left( a – 2b – 3c \right)^{2} = a^{2} + \left( - 2b \right)^{2} + \left( - 3c \right)^{2} + 2\left( a \right)\left( - 2b \right) + 2\left( - 2b \right)\left( - 3c \right) + 2\left( - 3c \right)\left( a \right)\]
By removing the parentheses and multiplying the terms,
We get,
\[\left( a – 2b – 3c \right)^{2} = \ a^{2} + 4b^{2} + 9c^{2} – 4ab + 12bc – 6ca\]
Thus the expansion of \[\left( a – 2b – 3c \right)^{2} = \ a^{2} + 4b^{2} + 9c^{2} – 4ab + 12bc – 6ca\]
Final answer :
The expansion of \[\left( a – 2b – 3c \right)^{2} = \ a^{2} + 4b^{2} + 9c^{2} – 4ab + 12bc – 6ca\] .

Note: Expanding algebraic expression is nothing but combining one or more variables or numbers by performing the given algebraic operations. We need to use distributive property to remove the parentheses. We can also expand this in another method also. That is by expanding the exponent into two terms then multiple the terms and then by simplifying we can expand the expression.
Additional information :
Another method :
Given , \[\left( a – 2b – 3c \right)^{2}\]
Expand the exponent,
\[\left( a – 2b – 3c \right)^{2} = \left( a – 2b – 3c \right)\left( a – 2b – 3c \right)\]
By distributing the terms,
\[a\left( a – 2b – 3c \right) – 2b\left( a – 2b – 3c \right) – 3c\left( a – 2b – 3c \right)\]
By removing the parentheses and by multiplying the terms,
We get,
\[\ a^{2}\ - 2ab – 3ac - \left( 2ba – 4b^{2} – 6bc \right)\left( 3ca – 6cb – 9c^{2} \right)\]
By removing the parentheses,
We get,
\[a^{2} – 2ab – 3ac – 2ba + 4b^{2} + 6bc – 3ca + 6cb + 9c^{2}\]
By collecting the like terms together
We get,
\[a^{2} + \left( - 2ab – 2ab \right) + \left( - 3ac – 3ac \right) + 4b^{2}\ + \left( 6bc + 6bc \right) + 9c^{2}\]
By adding the like terms together,
We get,
 \[a^{2} – 4ab – 6ac + 4b^{2} + 12bc + 9c^{2}\]
The expansion of \[\left( a – 2b – 3c \right)^{2} = \ a^{2} + 4b^{2} + 9c^{2} – 4ab + 12bc – 6ca\]
This is another method to expand the algebraic expression.