Question

How do you long divide `(4x^2-17x-11)/(7x+3)`?

Answer 1

`4/7x-107/49(-(218/49)/(7x+3))`
see long division solution method below

Warning: this is not pretty!

Explanation:

Write as a long division problem:
`color(white)("XXXX")underline(color(white)("XXXXXXXXXX")`
`7x+3 ) 4x^2-17x-11`

Divide the leading term of the divisor into the leading term of the dividend to get the first term of the quotient.

`color(white)("XXXXX")underline(color(red)(4/7x)color(white)("XXXXXXX")`
`color(red)(7x)+3) 4x^2-17x-11`

Multiply the entire divisor by the first term of the quotient
and subtract this from the original dividend.

`color(white)("XXXX")underline(color(red)(4/7x)color(white)("XXXXXXX")`
`color(red)(7x+3) ) 4x^2-17x-11`
`color(white)("XXXX")underline(color(red)(4x^2-12/7xcolor(white)("XXX")))`
`color(white)("XXXXXXX")color(blue)(-107/7x-11)`

Divide the first term of the divisor into the first term of the new dividend/remainder to get the second term of the quotient.

`color(white)("XXXX")underline(4/7xcolor(red)(-107/49)color(white)("XXX")`
`color(red)(7x)+3 ) 4x^2-17x-11`
`color(white)("XXXX")underline(4x^2-12/7xcolor(white)("XXX"))`
`color(white)("XXXXXXX")color(red)(-107/7x)-11`

Multiply the entire divisor by this second term in the quotient and subtract from the dividend/remainder.

`color(white)("XXXX")underline(4/7xcolor(red)(-107/49)color(white)("XXX")`
`color(red)(7x+3) ) 4x^2-17x-11`
`color(white)("XXXX")underline(4x^2-12/7xcolor(white)("XXX"))`
`color(white)("XXXXXXX")-107/7x-11`
`color(white)("XXXXXXX")underline(color(red)(-107/7x-321/49))`
`color(white)("XXXXXXXXXXXX")color(blue)(-218/49)`

`color(white)("XXXX")underline(4/7x-107/49color(white)("XXX")`(Quotient)
`7x+3 ) 4x^2-17x-11`
`color(white)("XXXX")underline(4x^2-12/7xcolor(white)("XXX"))`
`color(white)("XXXXXXX")-107/7x-11`
`color(white)("XXXXXXX")underline(-107/7x-321/49)`
`color(white)("XXXXXXXXXXXX")-218/49`(Remainder)