please help me solve thisView attachment 65087

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please help me solve thisView attachment 65087

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Let f(x) = kx³ + 5x² + 10x + 4

According to factor theorem, f(-1 + √3 i) = 0

Thus,

k(-1 + √3 i)³ + 5(-1 + √3 i)² + 10(-1 + √3 i) + 4 = 0

Now solve for k,

k = 2

So, f(x) = 2x³ + 5x² + 10x + 4

Also, f(x) = {x - (-1 + √3 i)}{x - (-1 - √3 i)}{Ax + B}

f(x) = {x² + 2x + 4}{Ax + B} = Ax³ + (B + 2A)x² + (2B + 4A)x + 4B

Compare coefficient of x³: A = 2

Compare constant: B = 1

f(x) = (x² + 2x + 4)(2x + 1)

Thus, third root is -1/2.

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Your quadratic factor is incorrect. Should have +2x instead of - 2x thereHope it helps.View attachment 65094

Sorry, my mistake. ThanksYour quadratic factor is incorrect. Should have +2x instead of - 2x there

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