<s>[INST] The number of solutions to the equation \(x^{4}+3x^{3}-24x^{2}+18x+20=0\) is to be determined. The polynomial has real coefficients, and one real root, \(x=-2\), is known. By the Factor Theorem, \(x+2\) is a factor of \(x^{4}+3x^{3}-24x^{2}+18x+20\). Dividing \(x^{4}+3x^{3}-24x^{2}+18x+20\) by \(x+2\) gives \(x^{3}-x^{2}-26x+10\). The quotient is a third degree polynomial with real coefficients. Therefore, one of the following is true: (1) The remaining roots are all real, or (2) One of the remaining roots is real and the other two are complex conjugates. Case (1): If all roots are real, then the cubic polynomial \(x^{3}-x^{2}-26x+10\) must have at least one real root. Note that \((x-1)(x+2)(x+5)=x^{3}+6x^{2}-3x-10\). By adding \(-7x^{2}-23x+20\) to this equation, \(x^{3}-x^{2}-26x+10\) is obtained. Therefore, \(x^{3}-x^{2}-26x+10=(x-1)(x+2)(x+5)-7x^{2}-23x+20\). If \(x>1\), then \((x-1)(x+2)(x+5)-7x^{2}-23x+20>0\) since \((x-1)(x+2)(x+5)>7x^{2}+23x-20\) when \(x>1\). If \(x<-5\), then \((x-1)(x+2)(x+5)-7x^{2}-23x+20>0\) since \((x-1)(x+2)(x+5)<-7x^{2}-23x+20\) when \(x<-5\). Note that \((x-1)(x+2)(x+5)-7x^{2}-23x+20=0\) when \(x=1\), and \((x-1)(x+2)(x+5)-7x^{2}-23x+20=0\) when \(x=-5\). Since \(x^{3}-x^{2}-26x+10\) has a root of 1 and \(-5\), it must be the case that it has no other real root, since a real root of multiplicity 3 would cause the polynomial to be 0 everywhere, and two real roots would cause the graph to cross the \(x\)-axis at two different points, and there would be no room for the third real root. Therefore, the original equation must have exactly one real root of multiplicity 1, one real root of multiplicity 1, and two complex conjugate roots. Case (2) Therefore, if the original equation has four real roots, then it must have exactly one real root of multiplicity 1, and three real roots of multiplicity 1. Therefore, the original equation must have at least two real roots of multiplicity 1, and the original equation cannot have more than two real roots of multiplicity 1. Therefore, the original equation must have exactly two real roots of multiplicity 1, and one pair of complex conjugate roots.[/INST]