Question

x^4+4(Factorise)
Its answer is(x^2+2x+2)(x^2-2x+2)
But don't verify only Factorise

Answer 1

$\textbf{ \huge{ Hello Mate! }}$

${x}^{4} + 4 = {x}^{4} + {2}^{2}$

Now, we have to write it in a form where such how it matches the identities.

${a}^{2} + { b }^{2} = {(a + b)}^{2} - 2ab$

So we can write it as,

${x}^{4} + {2}^{2} = {( {x}^{2} + 2) }^{2} - 2 {x}^{2} \times 2 \\ = {( {x}^{2} + 2)}^{2} - {(2x)}^{2}$
This meets our identity,

${a}^{2} - {b}^{2} = (a + b)(a - b)$

Hence,

$= ( {x}^{2} + 2 - 2x)( {x}^{2} + 2 + 2x)$

I hope I was succeful in soughting out you problem.

$\textbf{ Have great future ahead! }$

Answer 2

Method - 1:

Given Equation is x^4 + 4

Add and subtract 4x^2,we get

⇒ x^4 + 4 + 4x^2 - 4x^2

⇒ (x^4 + 4 + 4x^2) - 4x^2

⇒ (x^2 + 2)^2 - (2x)^2

We know that a^2 - b^2 = (a + b)(a - b)

⇒ (x^2 + 2 + 2x)(x^2 + 2 - 2x)

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Method - 2:

Given Equation is x^4 + 4.

It can be written as:

⇒ x^4 - 2x^3 + 2x^3 + 4x^2 - 2x^2 - 2x^2 + 4x - 4x + 4

It can be rearranged as

⇒ x^4 - 2x^3 + 2x^2 + 2x^3 - 4x^2 + 4x + 2x^2 - 4x + 4

⇒ x^2(x^2 - 2x + 2) + 2x(x^2 - 2x + 2) + 2(x^2 - 2x + 2)

⇒ (x^2 + 2x + 2)(x^2 - 2x + 2).

Hope it helps!