Factorize the polynomial:
\[4{a^2} - \left( {4{b^2} + 4bc + {c^2}} \right)\]

Hint: In this question, we have to factorize the given expression.
We have to factorize the given expression. For that first, we need to use the algebraic formula for the expression in the first bracket. Then again using another algebraic formula we will get the required result.
Formula:
Algebraic formula:
\[{\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}\]
\[{a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)\]

Complete step-by-step solution:
It is given that, \[4{a^2} - \left( {4{b^2} + 4bc + {c^2}} \right)\]
We need to factorize the given expression.
For factorization, we first need to factorize the term in the first bracket.
We know that, \[{\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}\]
We can simplify the expression \[\left( {4{b^2} + 4bc + {c^2}} \right)\] as \[\left( {\left( {2b} \right)^2} + 2 \times 2b \times c + {c^2}\right)\],
Now if we compare this with \[{\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}\], we can write \[a = 2b\] and \[b = c\].
Then, \[{\left( {2b} \right)^2} + 2 \times 2b \times c + {c^2}\] can be written as \[{(2b + c)^2}\].
Thus, we get,
\[4{a^2} - \left( {4{b^2} + 4bc + {c^2}} \right)\]
\[ = 4{a^2} - {\left( {2b + c} \right)^2}\]
\[ = {\left( {2a} \right)^2} - {\left( {2b + c} \right)^2}\]
\[ = \left( {2a + 2b + c} \right)\left\{ {2a - \left( {2b + c} \right)} \right\}\] [Using the formula, \[{a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)\]]
Simplifying we get,
\[ = \left( {2a + 2b + c} \right)\left( {2a - 2b - c} \right)\]
Hence, by factoring \[4{a^2} - \left( {4{b^2} + 4bc + {c^2}} \right)\] we get \[\left( {2a + 2b + c} \right)\left( {2a - 2b - c} \right)\].
Additional information:
Factor theorem:
In algebra, the factor theorem is a theorem linking factors and zeros of a polynomial.
The factor theorem will state that a polynomial \[f(x)\] has a factor \[\left( {x - a} \right)\] if and only if \[f\left( a \right) = 0\]
 (i.e. a is a root).
i.e. let,\[x = a\] is a solution of \[f(x)\], that could make \[\left( {x - a} \right)\] a factor.

Note: Algebraic expression:
In mathematics, an algebraic expression is an expression built up from integer constants, variables, and algebraic operations (addition, subtraction, multiplication, division, and exponentiation by an exponent that is a rational number).
For example, \[{x^2} + 6xy + 7\] is an algebraic expression where \[7\] is the integer constants and x and y
are the variables, + is the algebraic operations.
Factorization:
In mathematics, factorization means writing a number or algebraic expression as a product of several factors, mostly simpler and the same kind of factors.