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A factored polynomial reveals its roots, a key concept in understanding the behavior of these expressions.

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The function \(f(x)=x^4+ax^3+bx^2+cx+d\) has four positive roots. We are also given that \((b-a)(a+b)+(d-c)(c+d)=2(ac-bd).\)

What is the value of \((a+b+c+d)^2?\)

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Let \(f(x) = x^3 + 2x^2 + 3x + 2\) and \(g(x)\) be a polynomial with integer coefficients. When \(f(x)\) is divided by \(g(x)\), it leaves quotient \(q(x)\) and remainder \(r(x)\).

If \(q(x) = r(x) \neq 1\) ,then find \(g(5)\).

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\[\large x^6 + x^5 + 2x^4 + 2x^3 + 3x^2 + 3x + 3 \]

How many real roots does the polynomial above have?

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