Step 1

\(p(x)=x^{4}+x^{3}-8x^{2}-2x+12\)

\(\Rightarrow p(x)=x^{3}(x-2)+3x^{2}(x-2)-2x(x-2)-6(x-2)\)

\(\Rightarrow p(x)=(x-2)(x^{3}+3x^{2}-2x-6)\)

\(\Rightarrow p(x)=(x-2)[x^{2}(x+3)-2(x+3)]\)

\(\Rightarrow p(x)=(x-2)(x+3)(x^{2}-2)\)

\(\Rightarrow p(x)=(x-2)(x+3)(x+\sqrt{2})(x-\sqrt{2})\)

Step 2

linear factorization of p(x).

\(p(x)=(x-2)(x+3)(x+\sqrt{2})(x-\sqrt{2})\)

Zeros of p(x):

\(2,-3,\sqrt{2},-\sqrt{2}\).

\(p(x)=x^{4}+x^{3}-8x^{2}-2x+12\)

\(\Rightarrow p(x)=x^{3}(x-2)+3x^{2}(x-2)-2x(x-2)-6(x-2)\)

\(\Rightarrow p(x)=(x-2)(x^{3}+3x^{2}-2x-6)\)

\(\Rightarrow p(x)=(x-2)[x^{2}(x+3)-2(x+3)]\)

\(\Rightarrow p(x)=(x-2)(x+3)(x^{2}-2)\)

\(\Rightarrow p(x)=(x-2)(x+3)(x+\sqrt{2})(x-\sqrt{2})\)

Step 2

linear factorization of p(x).

\(p(x)=(x-2)(x+3)(x+\sqrt{2})(x-\sqrt{2})\)

Zeros of p(x):

\(2,-3,\sqrt{2},-\sqrt{2}\).