Given $\alpha$ and $\beta$ are the roots of the quadratic equation ${x}^{2}-4x+k=0(k\neq 0)$. If $ {\alpha \beta}^{2}+{\alpha}^{2}\beta$.

Using the relationship between roots and coefficients for x^2 - 4x + k = 0, we know the sum of the roots alpha + beta equals 4 and the product alpha * beta equals k. The given expression alpha * beta^2 + alpha^2 * beta factors to alpha * beta multiplied by (alpha + beta). Substituting the known values gives k multiplied by 4, which equals 4k.

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