Short Fall 2023 Question 10, 4.3.39 Part 1 of 2 HW Score: $75 \%, 7.5$ of 10 points Points: 0 of 1 Factor the polynomial function $f(x)$. Then solve the equation $f(x)=0$. \[ f(x)=x^{3}+6 x^{2}+3 x-10 \] The factored polynomial function is $f(x)=$ (Factor completely)

Step 1 ：The first step to solve this problem is to factor the polynomial function. Factoring is the process of breaking down a polynomial into simpler terms (the factors) that, when multiplied together, equal the original polynomial. This can be done by finding the roots of the polynomial, which are the values of x that make the polynomial equal to zero. Once we have the roots, we can express the polynomial as a product of factors of the form (x - root).

Step 2 ：The factored form of the polynomial is \((x - 1)(x + 2)(x + 5)\) and the roots of the polynomial are \(-5, -2, 1\). These are the values of \(x\) that make the polynomial equal to zero.

Step 3 ：Final Answer: The factored polynomial function is \(f(x) = (x - 1)(x + 2)(x + 5)\) and the solutions to the equation \(f(x) = 0\) are \(x = -5, -2, 1\).