Question

Factorise \[4{a^2} - 4ab + {b^2} - 9{c^2} + 12cd + 4{d^2}\],
\[A)(2a - b - 3c - 2d)(2a + b + 3c + 2d)\]
\[B)(2a - b + 3c + 2d)(2a + b - 3c - 2d)\]
\[C)(2a - b + 3c - 2d)(2a - b - 3c + 2d)\]
\[D)(2a + b + 3c - 2d)(2a - b + 3c - 2d)\]

Answer 1

Hint: In this question we need to factorize the polynomial expression and find the factors of the given polynomial. Factorise means factors of the given number or variable or expression. Here we use formula to factorise since it is not a quadratic expression, we are familiar with quadratic expression factorization. We split the given expression into quadratic expressions and then with the required algebraic formula we have to factorise in quadratic factorisation method and complete step by step explanation.

Formula used: \[{a^2} - 2ab + {b^2} = {(a + b)^2}\]
${a^2} - {b^2} = (a - b)(a + b)$

Complete step-by-step solution:
First consider given polynomial expression $4{a^2} - 4ab + {b^2} - 9{c^2} + 12cd + 4{d^2}$
It looks we can solve by quadratic expression
Now, we split the above expression in the form quadratic expression
$\Rightarrow$$(4{a^2} - 4ab + {b^2}) - (9{c^2} - 12cd - 4{d^2})$
Now, it looks like the formula mentioned in formula used and here we use the formula
$\Rightarrow$\[[{(2a)^2} - 2(2a)(b) + {b^2}] - [(3c) - 2(3c)(2d) + {(2d)^2}]\]
Now, we use formula mentioned in formula used, we get
$\Rightarrow$${(2a - b)^2} - {(3c - 2d)^2}$
By applying formula mentioned in formula used we, get
$\Rightarrow$\[(2a - b + 3c - 2d)(2a - b - (3c - 2d))\]
By applying basic subtraction, we get
$\Rightarrow$\[(2a - b + 3c - 2d)(2a - b - 3c + 2d)\]
Here obtained the solution required from the option given above

Option C is the correct answer.

Note: In mathematics, an algebraic expression is an expression built up from integer constants, variables, and the algebraic operations (addition, subtraction, multiplication, division and exponentiation by exponent that is a rational number). Transcendental numbers like $\pi $ and $e$ are not algebraic, since they are not derived from integer constants and algebraic operations. Usually, Pi is constructed as a geometric relationship, and the definition of $e$ requires an infinite number of algebraic operations.