Question

Factories: $(x^2-4x)(x^2-4x-1)-20$

Answer 1

Given: $(x^2-4x)(x^2-4x-1)-20$

Let $P=(x^2-4x)$

$\therefore (x^2-4x)(x^2-4x-1)-20=P(P-1)-20$

$\Rightarrow P^2-P-20$

Now, two numbers whose product is $-20P^2$ and whose sum is $-P$ are $-5P$ and $4P$.

Therefore, using Middle Term Splitting, we get,

$\Rightarrow P^2-5P+4P-20$

$\Rightarrow P(P-5)+4(P-20)$

$\Rightarrow (P-5)(P+4)$

$\Rightarrow (x^2-4x-5)(x^2-4x+4)$ ...(1)

Now, consider $x^2-4x-5$. Using, middle term splitting, we get,

$\Rightarrow x^2-5x+x-5$

$\Rightarrow x(x-5)+1(x-5)$

$\Rightarrow (x-5)(x+1)$

$\therefore x^2-4x-5=(x-5)(x+1)$ ...(2)

Now, consider $x^2-4x+4$

$\Rightarrow (x{)}^2+(2{)}^2-2\times 2\times x$

$\Rightarrow (x-2{)}^2$

$\therefore x^2-4x+4=(x-2{)}^2$ ...(3)

Hence, from (i), (2) and (3), we get,

$(x^2-4x)(x^2-4x-1)-20=(x-5)(x+1)(x-2{)}^2$

Hence, factorized.