lean_workbook_plus_9467

Let $f(x)=x^5+6x^4+7x^3-20x^2-42x-20$ . Note that $f(2)=32+96+56-80-84-20=0$ , so $2$ is a root of the polynomial. Factoring an $x-2$ out of the polynomial, we have $f(x)=(x-2)(x^4+8x^3+23x^2+26x+10)$ . Note that $-1$ is a root of the quartic in the factorization, so we can factor an $x+1$ out of the quartic to get $f(x)=(x-2)(x+1)(x^3+7x^2+16x+10)$ . Now, note that $-1$ is a root of the cubic, so we can once more factor an $x+1$ out of the expression to get $f(x)=(x-2)(x+1)^2(x^2+6x+10)$ . The quadratic is $(x+3)^2+1$ , and thus does not have any real roots, and cannot be factored in the real numbers. Thus, the polynomial is $\boxed{(x-2)(x+1)^2(x^2+6x+10)}$ .