5. $$ 4 a^{3} b-8 a^{2} b^{2}+2 a b^{3} $$

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1. Identify the common factors in every term of
$$
4a^3b-8a^2b^2+2ab^3.
$$
Each term contains the factors $$ a $$ and $$ b $$, and the coefficients $$4$$, $$-8$$, and $$2$$ have a common factor $$2$$. Thus, the greatest common factor is
$$
2ab.
$$

2. Factor $$2ab$$ out of the expression:
$$
4a^3b-8a^2b^2+2ab^3 = 2ab\left( \frac{4a^3b}{2ab} - \frac{8a^2b^2}{2ab} + \frac{2ab^3}{2ab} \right).
$$
Simplify each term inside the parentheses:
$$
\frac{4a^3b}{2ab} = 2a^2,\quad \frac{8a^2b^2}{2ab} = 4ab,\quad \frac{2ab^3}{2ab} = b^2.
$$
So the expression becomes:
$$
2ab\left(2a^2 - 4ab + b^2\right).
$$

3. The factored form of the original expression is:
$$
2ab\left(2a^2 - 4ab + b^2\right).
$$

This is the final factorization over the integers.
The expression $$4a^{3}b - 8a^{2}b^{2} + 2ab^{3}$$ can be factored as $$2ab(2a^{2} - 4ab + b^{2})$$.

5. \( 4 a^{3} b-8 a^{2} b^{2}+2 a b^{3} \)

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