Question

Please help, I have attached a picture of the question! Thank you!

Answer 1

Factors:

A.
`x+3`

B.
`4x+3`

C.
`x-5`

Explanation:

Given polynomial:

`4x^4-21x^3-46x^2+219x+180`

Rewrite it as follows:

`4x^4-21x^3-46x^2+219x+180\\ \\=4x^4+12x^3-33x^3-99x^2+53x^2+159x+60x+180\\ \\=4x^3(x+3)-33x^2(x+3)+53x(x+3)+60(x+3)\\ \\=(x+3)(4x^3-33x^2+53x+60)`

Now factor polynomial
`4x^3-33x^2+53x+60:`

`4x^3-33x^2+53x+60\\ \\=4x^3-16x^2-17x^2+68x-15x+60\\ \\=4x^2(x-4)-17x(x-4)-15(x-4)\\ \\=(x-4)(4x^2-17x-15)`

Now factor polynomial
`4x^2-17x-15:`

`4x^2-17x-15\\ \\=4x^2-20x+3x-15\\ \\=4x(x-5)+3(x-5)\\ \\=(x-5)(4x+3)`

Hence,

`4x^4-21x^3-46x^2+219x+180=(x+3)(x-4)(x-5)(4x+3)`