1. **State the problem:** Simplify the expression $$2(x-5)^2 + 4(x-5) + 4$$ into a polynomial in standard form. 2. **Recall the formula:** The square of a binomial is given by $$ (a-b)^2 = a^2 - 2ab + b^2 $$. 3. **Expand the squared term:** $$ 2(x-5)^2 = 2(x^2 - 2 \cdot x \cdot 5 + 5^2) = 2(x^2 - 10x + 25) $$ 4. **Distribute the 2:** $$ 2x^2 - 20x + 50 $$ 5. **Expand the linear term:** $$ 4(x-5) = 4x - 20 $$ 6. **Rewrite the entire expression:** $$ 2x^2 - 20x + 50 + 4x - 20 + 4 $$ 7. **Combine like terms:** - Combine the $$x$$ terms: $$-20x + 4x = -16x$$ - Combine the constants: $$50 - 20 + 4 = 34$$ 8. **Final simplified polynomial in standard form:** $$ \boxed{2x^2 - 16x + 34} $$