1. **Problem 1: Factor completely** the expression $$25x^2 - 9$$. 2. This is a difference of squares since $$25x^2 = (5x)^2$$ and $$9 = 3^2$$. 3. The formula for difference of squares is $$a^2 - b^2 = (a - b)(a + b)$$. 4. Applying the formula: $$25x^2 - 9 = (5x)^2 - 3^2 = (5x - 3)(5x + 3)$$. --- 5. **Problem 2: Divide the polynomials** $$\frac{2x^3 - 3x^2 - 2x + 3}{x - 1}$$ using polynomial long division. 6. Set up the division: dividend $$2x^3 - 3x^2 - 2x + 3$$ and divisor $$x - 1$$. 7. Divide the leading term of the dividend $$2x^3$$ by the leading term of the divisor $$x$$: $$\frac{2x^3}{x} = 2x^2$$. 8. Multiply the divisor by $$2x^2$$: $$2x^2(x - 1) = 2x^3 - 2x^2$$. 9. Subtract this from the dividend: $$\left(2x^3 - 3x^2\right) - \left(2x^3 - 2x^2\right) = -x^2$$. 10. Bring down the next term $$-2x$$, new expression: $$-x^2 - 2x$$. 11. Divide the leading term $$-x^2$$ by $$x$$: $$\frac{-x^2}{x} = -x$$. 12. Multiply divisor by $$-x$$: $$-x(x - 1) = -x^2 + x$$. 13. Subtract: $$\left(-x^2 - 2x\right) - \left(-x^2 + x\right) = -3x$$. 14. Bring down the next term $$+3$$, new expression: $$-3x + 3$$. 15. Divide leading term $$-3x$$ by $$x$$: $$\frac{-3x}{x} = -3$$. 16. Multiply divisor by $$-3$$: $$-3(x - 1) = -3x + 3$$. 17. Subtract: $$\left(-3x + 3\right) - \left(-3x + 3\right) = 0$$ remainder. 18. The quotient is: $$2x^2 - x - 3$$. --- **Final answers:** - Factorization: $$25x^2 - 9 = (5x - 3)(5x + 3)$$. - Division result: $$\frac{2x^3 - 3x^2 - 2x + 3}{x - 1} = 2x^2 - x - 3$$ with remainder 0.