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Factorize ${(x^2-4x)}{(x^2-4x-1)}-20$

${(x^2-4x)}{(x^2-4x-1)}-20$ is an algebraic expression in terms of $x$. In this problem, we have to transform this algebraic expression in product form by factorization.

Simplify the Algebraic expression

${(x^2-4x)}{(x^2-4x-1)}-20$ $\,=\,$ ${(x^2-4x)}{((x^2-4x)-1)}-20$

Now, multiply each term in the second factor by the first factor in the first term of the algebraic expression.

$= \,\,\,$ ${(x^2-4x)}{(x^2-4x)}$ $-$ $(x^2-4x) \times 1$ $-$ $20$

$= \,\,\,$ ${(x^2-4x)}^2$ $-$ $(x^2-4x)$ $-$ $20$

Convert expression in terms of another variable

The simplified algebraic expression ${(x^2-4x)}^2$ $-$ $(x^2-4x)$ $-$ $20$ can be expressed as a quadratic polynomial. In order to factorize this algebraic expression, take $u = x^2-4x$ and convert the quadratic expression in terms of $u$.

$\implies$ ${(x^2-4x)}^2$ $-$ $(x^2-4x)$ $-$ $20$ $\,=\,$ $u^2-u-20$

Factorize the Quadratic expression

Look at the third term in the quadratic expression $u^2-u-20$, the term $20$ can be factored in three ways by factorization.

$(1) \,\,\,$ $20 = 1 \times 20$
$(2) \,\,\,$ $20 = 2 \times 10$
$(3) \,\,\,$ $20 = 4 \times 5$

Choose a right combination of factoring to express the third term as product of two quantities and second term as either sum or difference form in the same combination.

$\implies$ $u^2-u-20$ $\,=\,$ $u^2-5u+4u-5 \times 4$

$u$ is a common factor in the first two terms and $4$ is a common factor in the remaining two terms. So, take each factor common from them.

$= \,\,\,$ $u(u-5)+4(u-5)$

Now, $u-5$ is a common factor in the both terms. So, take it common from them.

$= \,\,\,$ $(u-5)(u+4)$

Actually, the algebraic expression is not in terms of $u$. So, bring this algebraic expression back to original form by replacing $u$ by $x^2-4x$.

$\therefore \,\,\,$ $(u-5)(u+4)$ $\,=\,$ $(x^2-4x-5)(x^2-4x+4)$

Therefore, the algebraic expression ${(x^2-4x)}^2$ $-$ $(x^2-4x)$ $-$ $20$ is factored as $(x^2-4x-5)(x^2-4x+4)$ by factorisation.

Factorize each Quadratic expression


The first factor in the product can be factored by the factoring method and the second factor can be simplified as per square of the difference formula.

$= \,\,\,$ $(x^2-5x+x-5)$ $(x^2-2 \times x \times 2+2^2)$

$x$ is a common factor in the first two terms and $1$ is common factor in the next two terms in the first factor of the algebraic expression. The second factor can be simplified as square of the difference of two terms.

$= \,\,\,$ $(x(x-5)+1(x-5)){(x-2)}^2$

Now, $x-5$ is a common factor in the both terms of the first factor of the algebraic expression. Take $x-5$ common from them and simplify the algebraic expression in factoring form.

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

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