Question

$( {a}^{2} - 3a)( {a}^{2} - 3a + 7) + 10$
Factorise please brothers and sisters
Urgent!!!​

Answer 1

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

Given expression is

$\rm :\longmapsto\:( {a}^{2} - 3a)( {a}^{2} - 3a + 7) + 10$

➢ Let assume that

$\red{\rm :\longmapsto\:{a}^{2} - 3a = x}$

So, given expression can be rewritten as

$\rm \:  =  \:\:x(x + 7) + 10$

$\rm \:  =  \:  \: {x}^{2} + 7x + 10$

$\rm \:  =  \:  \: {x}^{2} + 5x + 2x + 10$

$\rm \:  =  \:  \:x(x + 5) + 2(x + 5)$

$\rm \:  =  \:  \:(x + 5)(x + 2)$

$\rm \:  =  \:  \:( {a}^{2} - 3a + 5)( {a}^{2} - 3a+ 2)$

$\rm \:  =  \:  \:( {a}^{2} - 3a + 5)( {a}^{2} - 2a - a+ 2)$

$\rm \:  =  \:  \:( {a}^{2} - 3a + 5)\bigg(a(a - 2) - 1(a - 2)\bigg)$

$\rm \:  =  \:  \:( {a}^{2} - 3a + 5)\bigg((a - 2)(a - 1)\bigg)$

$\rm \:  =  \:  \:( {a}^{2} - 3a + 5)(a - 2)(a - 1)$

Hence,

$\bf :\longmapsto\:( {a}^{2} - 3a)( {a}^{2} - 3a + 7) + 10$

$\bf \:  =  \:  \:( {a}^{2} - 3a + 5)(a - 2)(a - 1)$

Additional Information :-

Let's solve one more problem of same type!!

Question :- Factorize the following :-

$\rm :\longmapsto\:( {a}^{2} - 4a)( {a}^{2} - 4a - 1) - 20$

➢ Let assume that

$\red{\rm :\longmapsto\:{a}^{2} - 4a = x}$

So, above expression can be rewritten as

$\rm \:  =  \: \:x(x - 1) - 20$

$\rm \:  =  \: \: {x}^{2} - x - 20$

$\rm \:  =  \: \: {x}^{2} - 5x + 4x - 20$

$\rm \:  =  \:  \:x(x - 5) + 4(x - 5)$

$\rm \:  =  \:  \:(x - 5)(x + 4)$

$\rm \:  =  \:  \:( {a}^{2} - 4a- 5)( {a}^{2} - 4a + 4)$

$\rm \:  =  \:  \:( {a}^{2} - 5a + a- 5)( {a}^{2} - 2a - 2a + 4)$

$\rm \:  =  \:  \:\bigg(a(a - 5) + 1(a - 5)\bigg)\bigg(a(a - 2) - 2(a - 2)\bigg)$

$\rm \:  =  \:  \:(a - 5)(a + 1)(a - 2)(a - 2)$

Hence,

$\bf :\longmapsto\:( {a}^{2} - 4a)( {a}^{2} - 4a - 1) - 20$

$\bf \:  =  \:  \:(a - 5)(a + 1)(a - 2)(a - 2)$