<s>[INST] Find a polynomial of degree $4$ with the coefficient $x^2$ equal to $6$, and zeros $-3$, $ 2$, $ -1$, $-2$ I started off with:
$$f(x)= a(x-(-3)) (x-(2)) (x-(-1)) (x-(-2))$$
$$f(x)= a(x+3) (x-2) (x+1) (x+2)$$
 [/INST] You are right that $f(x)=a(x+3)(x+2)(x+1)(x-2)$
Expanding gives $$f(x)=a(x^{2}+3x+2x+6)(x^{2}+x-2x-2)=a(x^{2}+5x+6)(x^{2}-x-2)$$  
Multiplying the quadratics, $$f(x)=a(x^{4}+5x^{3}+6x^{2}-x^{3}-5x^{2}-6x-2x^{2}-10x-12)$$  
Collecting terms gives $f(x)=a(x^{4}+4x^{3}-x^{2}-16x-12)$
Hence, the coefficient on $x^{2}$ is $-a$  
To make the coefficient on $x^{2}$ $6$, let $a=-6$.
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