Question

factor 4 a^2 + 12 a b + 9 b^2 - 8 a - 12 b​

Answer 1

$\large\underline{\sf{Solution-}}$

Given expression is

$\rm :\longmapsto\: {4a}^{2} - 12ab + {9b}^{2} - 8a - 12b$

can be regrouped as

$\rm \:  =  \: \: ({4a}^{2} - 12ab + {9b}^{2}) - (8a + 12b)$

$\rm=\bigg(2a \times 2a + 12ab + 3b \times 3b\bigg) - (8a + 12b)$

$\rm \:  =\bigg( {(2a)}^{2} + 12ab + {(3b)}^{2} \bigg) - (2.2.2a + 2.2.3b)$

$\rm \:  =\bigg( {(2a)}^{2} + 2.2.3a.b + {(3b)}^{2} \bigg) - (2.2.2a + 2.2.3b)$

$\rm \:  =\bigg( {(2a)}^{2} + 2(2a)(3b) + {(3b)}^{2} \bigg) - (2.2.2a + 2.2.3b)$

We know,

$\red{ \boxed{ \sf{ \: {x}^{2} + {y}^{2} + 2xy = {(x + y)}^{2}}}}$

So, using this, we get

$\rm \:  =  \: {(2a + 3b)}^{2} - 4(2a + 3b)$

$\rm \:  =  \: {(2a + 3b)}(2a + 3b) - 4(2a + 3b)$

$\rm \:  =  \: {(2a + 3b)}\bigg[2a + 3b- 4\bigg]$

Concept Used :-

1. Factorisation by Regrouping Terms

➣Sometimes it happens that there is no common term in the expressions then

➣We have to make the groups of the terms.

➣Then choose the common factor among these groups.

➣Find the common factor and it will give the required factors.

Additional Information :-

1. Method of Common Factors

In this method, we have to write the irreducible factors of all the terms

Then find the common factors amongst all the irreducible factors.

The required factor form is the product of the common term and the left over terms is the required factorization.