Question : What is the LCM of $\left(8 x^3+80 x^2+200 x\right)$ and $\left(4 x^4+16 x^3-20 x^2\right)$?
Option 1: $8 x^2(x+5)^2(x-1)$
Option 2: $8 x^2(x-1)^2(x+5)$
Option 3: $4 x^2(x-1)^2(x+5)$
Option 4: $4 x^2(x+5)^2(x-1)$
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Correct Answer: $8 x^2(x+5)^2(x-1)$
Solution : Consider, $(8 x^3+80 x^2+200 x)$ Taking $8x$ as common, we get, $=8x(x^2+10x+25)$ $=8x(x^2+5x+5x+25)$ $=8x(x(x+5)+5(x+5))$ $=8x(x+5)^2$ Now consider, $(4 x^4+16 x^3-20 x^2)$ Taking $4x^2$ as common, we get, $=4x^2(x^2 + 4x-5)$ $=4x^2(x^2-x+5x-5)$ $=4x^2(x(x-1)+5(x-1))$ $=4x^2(x-1)(x+5)$ $\therefore$ LCM $=8x^2(x+5)^2(x-1)$ Hence, the correct answer is $8x^2(x+5)^2(x-1)$.
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