To find the roots of the polynomial equation [tex]\( x^3 - 10x = -3x^2 + 24 \)[/tex], we first need to bring all terms to one side of the equation to set it equal to zero.
Starting with the given equation:
[tex]\[ x^3 - 10x = -3x^2 + 24 \][/tex]
Let's move all terms to the left-hand side:
[tex]\[ x^3 - 10x + 3x^2 - 24 = 0 \][/tex]
Next, we combine like terms:
[tex]\[ x^3 + 3x^2 - 10x - 24 = 0 \][/tex]
Now, our task is to find the roots of this polynomial equation:
[tex]\[ x^3 + 3x^2 - 10x - 24 = 0 \][/tex]
We can approach this problem by trying potential rational roots using the Rational Root Theorem, but in this scenario, using a graphing calculator or system of equations would be more efficient. From the result, we know the roots of the polynomial are:
[tex]\[ x = -4 \][/tex]
[tex]\[ x = -2 \][/tex]
[tex]\[ x = 3 \][/tex]
Therefore, the roots of the polynomial [tex]\( x^3 + 3x^2 - 10x - 24 = 0 \)[/tex] are [tex]\(-4\)[/tex], [tex]\(-2\)[/tex], and [tex]\(3\)[/tex]. Thus, the correct solution from the provided choices is:
[tex]\[ \boxed{-4, -2, 3} \][/tex]
