ANSWER:
[tex]y=-\frac{x^{6}}{4}-\frac{x^{5}}{4}+\frac{25x^{4}}{4}+\frac{25x^{3}}{4}-45x^2-27x+108[/tex]
STEP-BY-STEP EXPLANATION:
Taking into account the above, we can establish that the equation of the polynomial is:
[tex]P(x)=-\frac{1}{4}\mleft(x+3\mright)^3\mleft(x-2\mright)^2\mleft(x-4\mright)[/tex]
We solve and we would be left as follows:
[tex]\begin{gathered} (x+3)^3(x-2)^2(x-4) \\ (x+3)^3=x^3+9x^2+27x+27 \\ (x-2)^2=x^2-4x+4 \\ \text{ we replacing} \\ (x^3+9x^2+27x+27)(x^2-4x+4)(x-4) \\ (x^2-4x+4)(x-4)=x^3+x^2\mleft(-4\mright)-4x^2-4x\mleft(-4\mright)+4x+4\mleft(-4\mright)=x^3-8x^2+20x-16 \\ (x^3+9x^2+27x+27)(x^3-8x^2+20x-16) \\ x^3x^3+x^3\mleft(-8x^2\mright)+x^3\cdot\: 20x+x^3\mleft(-16\mright)+9x^2x^3+9x^2\mleft(-8x^2\mright)+9x^2\cdot\: 20x+9x^2\mleft(-16\mright)+27xx^3+27x\mleft(-8x^2\mright)+27x\cdot\: 20x+27x\mleft(-16\mright)+27x^3+27\mleft(-8x^2\mright)+27\cdot\: 20x+27\mleft(-16\mright) \\ x^6-8x^5+20x^4-16^3+9x^5-72x^4+180x^3-144x^2+27x^4-216x^3+540x^2-432x+27x^3-216x^2+540x-432 \\ -\frac{1}{4}\cdot(x^6+x^5-25x^4-25x^3+180x^2+108x-432) \\ -\frac{x^6}{4}-\frac{x^5}{4}+\frac{25x^4}{4}+\frac{25x^3}{4}-45x^2-27x+108 \\ y=-\frac{x^{6}}{4}-\frac{x^{5}}{4}+\frac{25x^{4}}{4}+\frac{25x^{3}}{4}-45x^2-27x+108 \end{gathered}[/tex]
