1. Problem: Simplify the expression $\left(a-b\right)^2\left(a^2 + 4\right) + \left(a-b\right)\left(a^2 + b^2\right)$. 2. Formula and rules: Use distributive property and factor common terms. Recall that $\left(a-b\right)^2 = \left(a-b\right)\left(a-b\right)$. 3. Step 1: Factor out the common factor $\left(a-b\right)$ from both terms: $$\left(a-b\right)^2\left(a^2 + 4\right) + \left(a-b\right)\left(a^2 + b^2\right) = \left(a-b\right)\left[\left(a-b\right)\left(a^2 + 4\right) + \left(a^2 + b^2\right)\right]$$ 4. Step 2: Expand inside the bracket: $$\left(a-b\right)\left[\left(a-b\right)\left(a^2 + 4\right) + \left(a^2 + b^2\right)\right] = \left(a-b\right)\left[\left(a-b\right)a^2 + \left(a-b\right)4 + a^2 + b^2\right]$$ 5. Step 3: Distribute $\left(a-b\right)$ inside: $$= \left(a-b\right)\left[a^2 a - a^2 b + 4a - 4b + a^2 + b^2\right]$$ 6. Step 4: Combine like terms inside the bracket: $$= \left(a-b\right)\left[a^3 - a^2 b + 4a - 4b + a^2 + b^2\right]$$ 7. Step 5: Group terms for clarity: $$= \left(a-b\right)\left[a^3 + a^2 - a^2 b + b^2 + 4a - 4b\right]$$ This is the simplified form. Further factorization depends on the context but this is the expanded and factored form. Final answer: $$\boxed{\left(a-b\right)\left[a^3 + a^2 - a^2 b + b^2 + 4a - 4b\right]}$$