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Can Sagemath factor polynomial expressions with square root or in general any root?

For example this expression:

$$-x^3+5 x^2+3 x+3+\sqrt{x^2+1} \left(2 x^3+x^2+3\right)$$

-x^3 + 5*x^2 + 3*x + 3 + sqrt(x^2 + 1)*(2*x^3 + x^2 + 3)

The results should be in the form (assuming two factors):

$(f_1(x) + f_2(x)\sqrt{x^2+1}) * (g_1(x) + g_2(x)\sqrt{x^2+1})$

where f1(x), f2(x), g1(x), g2(x) are normal polynomials without any roots.

$\sqrt{x^2+1}$ is like an extension but I do not know the right terminology how to name it.

Can Sagemath factor polynomial expressions with square root or in general any root?

For example this expression:

$$-x^3+5 x^2+3 x+3+\sqrt{x^2+1} \left(2 x^3+x^2+3\right)$$

-x^3 + 5*x^2 + 3*x + 3 + sqrt(x^2 + 1)*(2*x^3 + x^2 + 3)

The results should be in the form (assuming two factors):

$(f_1(x) $$(f_1(x) + f_2(x)\sqrt{x^2+1}) * (g_1(x) + g_2(x)\sqrt{x^2+1})$\sqrt{x^2+1})$$

where f1(x), f2(x), g1(x), g2(x) are normal polynomials without any roots.

$\sqrt{x^2+1}$ is like an extension but I do not know the right terminology how to name it.

Can Sagemath factor polynomial expressions with square root or in general any root?

For example this expression:

$$-x^3+5 x^2+3 x+3+\sqrt{x^2+1} \left(2 x^3+x^2+3\right)$$

-x^3 + 5*x^2 + 3*x + 3 + sqrt(x^2 + 1)*(2*x^3 + x^2 + 3)

The results should be in the form (assuming two factors):

$$(f_1(x) (f1(x) + f_2(x)\sqrt{x^2+1}) f2(x)*sqrt(x^2+1)) * (g_1(x) (g1(x) + g_2(x)\sqrt{x^2+1})$$g2(x)*sqrt(x^2+1))

where f1(x), f2(x), g1(x), g2(x) are normal polynomials without any roots.

$\sqrt{x^2+1}$ is like an extension but I do not know the right terminology how to name it.