Question

Resolve into factors: $4m^4+9n^4-24m^2{n}^2$

Answer 1

In the given expression $4m^4+9n^4-24m^2{n}^2$, add and subtract $12m^2{n}^2$ to make it a perfect square as shown below:

$4m^4+9n^4-24m^2{n}^2=(4m^4+9n^4+12m^2{n}^2)-24m^2{n}^2-12m^2{n}^2$

$=\left[\right(2m^2{)}^2+(3n^2{)}^2+2(3\left)\right(2\left)m^2{n}^2\right]-24m^2{n}^2-12m^2{n}^2=(2m^2+3n^2{)}^2-36m^2{n}^2$

(using the identity $(a+b{)}^2=a^2+b^2+2ab$)

We also know the identity $a^2-b^2=(a+b)(a-b)$, therefore,

Using the above identity, the expression $(2m^2+3n^2{)}^2-36m^2{n}^2$can be factorised as follows:

$(2m^2+3n^2{)}^2-36m^2{n}^2=(2m^2+3n^2{)}^2-(6mn{)}^2=(2m^2+3n^2+6mn\left)\right(2m^2+3n^2-6mn)$

Hence, $4m^4+9n^4-24m^2{n}^2=(2m^2+3n^2+6mn)(2m^2+3n^2-6mn)$