Find the LCM?

y^2 - 9 = (y - 3)(y +3)

y^2 + 10y + 21 = ( y + 3)(y + 7)

y^2 + 4y - 21 = ( y - 3)(y + 7)

factor all of them and find the part that's in common.

y^2 - 9 = difference of squares = (y-3)(y+3)

y^2 + 10y + 21 = factor by grouping = (y+7)(y+3)

y^2 + 4y - 21 = factor by grouping = (y+7)(y-3)

So all three of these don't have 1 particular thing in common.. BUT between the first two equations, (y+3) is common, and between the last two equations, (y+7) is common, and between the outside equations (y-3) is common.

SO our lowest common multiple is (y+3)(y-3)(y+7)

y^2 - 9 = (y + 3)(y - 3)

y^2 + 10y + 21 = (y + 3)(y + 7)

y^2 + 4y - 21 = (y + 7)(y - 3)

LCM = (x - 3)(x + 3)(x + 7)

= (x^2 - 9)(x + 7)

= x^3 + 7x - 9x - 63

= x^3 - 2x - 63

y^2 - 9, y^2 + 10y + 21, y^2 + 4y - 21

(y + 3)(y - 3), (y + 3)(y + 7), (y + 3)(y - 7)

LCM:

(y + 3) (y - 3) (y + 7) (y - 7)

y^4 - 58 y^2 + 441

y^2 - 9 = (y + 3)(y - 3)

y^2 + 10y + 21 = (y + 3)(y + 7)

y^2 + 4y - 21 = (y + 7)(y - 3)

In order to be a multiple of all of these, it must contain all these factors that appear above: (y - 3), (y + 3), (y + 7). There aren't any others. Therefore if you multiply those three together you get the LCM of the three expresions.