MHB Solve Polynomial Division: -5a + 4b = x^2+1 Rem -A-2

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The remainder of the polynomial p(x) = x^3 + ax^2 + 4bx - 1 when divided by x^2 + 1 is -5a + 4b. Additionally, when p(x) is divided by x + 1, the remainder is -a - 2. To solve for the values of a and b, it is established that 4b - a + 1 = 0, indicating no x term remains. The next step involves substituting x = -1 to derive a second equation for further analysis. The goal is to find the value of 8ab based on these equations.
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The remainder of $$p(x)=x^3+ax^2+4bx-1$$ divided by $$x^2+1$$ is –5a + 4b. If the remainder of p(x) divided by x + 1 is –a – 2, the value of 8ab is ...
A. $$-\frac34$$
B. $$-\frac12$$
C. 0
D. 1
E. 3

Dividing p(x) by $$x^2+1$$ by $$x^2+1$$ with –5a + 4b as the remainder using long division, I got (4bx – 1) – ((a – 1)x + a – 1) = –5a + 4b, thus (4b – a + 1)x + a = –5a + 4b. Does this mean that 4b – a + 1 = 0 since the right hand doesn't have an x term? Or do I need to look for the value of x first? I'm at a loss here.
 
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yes. 4b – a + 1 = 0 as there is no x term.
you need to divide by x+1 and get the constant term (putting x = -1 shall do also)
this shall give 2 equations and you need to solve and proceed further;
 
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