Question

How do you factor `25 g^3 v^3 + 15 g^2 e^3 v^3 + 5 g b d + 3 e^3 b d`?

Answer 1

`25g^3v^3+15g^2e^3v^3+5gbd+3e^3bd=color(blue)((5g^2v^3+bd)(5g+3e^3))`

Explanation:

Note 1:
`color(white)("XXX")`The first two terms have a common factor of
`color(white)("XXXXXX")5g^2v^3`
`color(white)("XXX")25g^3v^3+15g^2e^3v^3=(color(red)(5g^2v^3))(5g+3e^3)`

Note 2:
`color(white)("XXX")`The last two terms have a common factor of
`color(white)("XXXXXX")bd`
`color(white)("XXX")5gbd+3e^3bd=(color(green)(bd))(5g+3e^3)`

Therefore:
`color(white)("XXX")25g^3v^3+15g^2e^3v^3+5gbd+3e^3bd`
`color(white)("XXXXXX")=(color(red)(5g^2v^3))(5g+3e^3)+(color(green)(bd))(5g+3e^3)`

Note 3:
`color(white)("XXX")`The second factor for each term is identical

`color(white)("XXX")25g^3v^3+15g^2e^3v^3+5gbd+3e^3bd`
`color(white)("XXXXXX")=(color(red)(5g^2v^3)+color(green)(bd))(5g+3e^3)`