The linear factorization of h(x), obtained using a Polynomial Roots Calculator is; (x - 2)·(x - 2)·(2·x - 1)·(x + 3) (none of the options are correct)
How the linear factors are found;
To perform the linear factorization of the polynomial h(x) = 2x⁴ − 3x³ − 15x² + 32x − 12, we need to find the factors of h(x) that are of the form (x − c), where c is a constant. One way to do this is to use the rational root theorem, which states that if h(x) has a rational root p/q, where p and q are integers with no common factors, then p is a factor of the constant term of h(x) and q is a factor of the leading coefficient of h(x).
In this case, the constant term of h(x) is -12 and the leading coefficient of h(x) is 2, so the possible rational roots of h(x) are ±1, ±2, ±3, ±4, ±6, ±12, ±1/2, ±3/2, and ±6/2. We can use synthetic division to test each of these candidates and see if they produce a zero remainder, which means they are roots of h(x). Alternatively, we can use a graphing calculator or an online tool to find the roots of h(x) graphically or numerically. Using ta roots calculator tool, we can find that the roots of h(x) are 2, 1/2, and -3. The number of roots indicate that one of the roots has a multiplicity of 2. Whereby the root with a multiplicity of 2 is; x = 2, we get;
(x - 2)·(x - 2)·(x + 3)·(2·x - 1) = 2·x⁴ - 3·x³ - 15·x² + 32·x - 12
Therefore, the linear factorization of h(x) is; (x - 2)·(x - 2)·(x + 3)·(2·x - 1)