1. **State the problem:** Simplify the expression $$3(x - 5)^2 - 2(x - 5) + 4$$ by distributing and writing it as 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. **Apply the formula:** $$ (x - 5)^2 = x^2 - 2 \cdot x \cdot 5 + 5^2 = x^2 - 10x + 25 $$ 4. **Substitute back:** $$ 3(x - 5)^2 - 2(x - 5) + 4 = 3(x^2 - 10x + 25) - 2(x - 5) + 4 $$ 5. **Distribute the coefficients:** $$ 3x^2 - 30x + 75 - 2x + 10 + 4 $$ 6. **Combine like terms:** $$ 3x^2 - 30x - 2x + 75 + 10 + 4 = 3x^2 - 32x + 89 $$ 7. **Final answer:** The simplified polynomial in standard form is $$\boxed{3x^2 - 32x + 89}$$.