Answer:
x^4 -12x^3 +36x^2 +32x -192
Explanation:
You want the LCM of (x^2 -4x -12) and (x^2 -8x +16).
LCM
The least common multiple is the product divided by the greatest common factor.
The factors are ...
x^2 -4x -12 = (x -6)(x +2)
x^2 -8x +16 = (x -4)^2
There are no common factors, so the LCM is the product of the two polynomials:
(x^2 -4x -12) × (x^2 -8x +16)
= (1·1)x^4 +(1·(-8) +(-4)·1)x^3 +(1·16 +1·(-12) +(-4)(-8))x^2 +(-4(16) +(-8)(-12))x +(-12)(16)
= x^4 -12x^3 +36x^2 +32x -192 . . . . LCM of the give polynomials
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Additional comment
You can use the distributive property 4 times to get the 9 product terms that need to be combined. Or, you can consider the coefficients that must be combined to give the coefficient of a given power of the product. That's what we did above.
If you have the coefficient arrays ...
a b c
d e g
for terms in decreasing-degree order, then the coefficients of the product in decreasing-degree order are ...
ad, ae+db, ag+dc+be, bg+ec, cg
If you look at this list carefully, you can see the pattern in the products. Once you know the pattern, you can usually do the arithmetic mentally.