Factorise $12 x y + 25 - 4 x^{2} - 9 y^{2}$ .

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Solution

Step I - Let's observe the given expression and rearrange them.
$12 x y + 25 - 4 x^{2} - 9 y^{2}$ = $25 - (4 x^{2} - 12 x y + 9 y^{2})$
Step II - As we know the algebraic identity,
${(a - b)}^{2} = a^{2} - 2 a b + b^{2}$
$25 - (4 x^{2} - 12 x y + 9 y^{2})$ = ${(5)}^{2} - {(2 x - 3 y)}^{2}$
Step III - Now we know the other algebraic identity.
$a^{2} - b^{2} = (a + b) (a - b)$
Here, we can factorise the above expression by using this identity.
${(5)}^{2} - {(2 x - 3 y)}^{2}$ = $(5 + 2 x - 3 y) (5 - 2 x + 3 y)$
Hence, the factors of $12 x y + 25 - 4 x^{2} - 9 y^{2}$ = $(5 + 2 x - 3 y) (5 - 2 x + 3 y)$ .

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