Question

Factor the following
(a²+b²)²-18(a²+b²)-88​

Answer 1

`\rm\displaystyle( {a}^(2) + {b}^(2) - 22)( {a}^(2) + {b}^(2) + 4)`

Explanation:

we would like to factor the following:

`\rm\displaystyle ( {a}^(2) + {b}^(2) {)}^(2) - 18( {a}^(2) + {b}^(2) ) - 88`

let a²+b²=x

thus substitute:

`\rm\displaystyle x {}^(2) - 18x- 88`

rewrite the middle term as 4x-22x:

`\rm\displaystyle x^(2) + 4x - 22x - 88`

factor out x:

`\rm\displaystyle x( x^{} + 4)- 22x - 88`

factor out -22:

`\rm\displaystyle x( x^{} + 4)- 22(x + 4)`

group:

`\rm\displaystyle( x- 22)(x + 4)`

substitute back:

`\rm\displaystyle( {a}^(2) + {b}^(2) - 22)( {a}^(2) + {b}^(2) + 4)`

and we are done!