<s>[INST] What number should be subtracted from What number should be subtracted from $4x^3+5x+3$
 so that the resulting polynomial leaves remainder $-80$ when divided by $2x+5$?.
Let the required number to be subtracted be $K$.
let: $$P(x)=4x^3+5x+3-k$$ 
$$g(x)=2x+5=2(x+5/2)$$
Comparing $g(x)$ with $x-a $ ,  $a =\frac {-5}{2}$
Solving then gives $K=8$
but the answer in my book is $-152$. how? 
 [/INST] Let $f (x)=2x+5$ and $p (x)=4x^3+5x-3$. We have that: $$p (-\frac {5}{2}) -k = -80$$ $$\Rightarrow 4 (-\frac {5}{2})^3 +5 (-\frac {5}{2}) +3 - k = -80$$ $$-75+3-k = -80$$ $$\boxed {k = 8} $$ Hope it helps. 
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