1. Simplify the expression $4x^2 + 3x - 5 - (-3x^2 - 8x + 2)$. Start by distributing the minus sign to the second polynomial: $$4x^2 + 3x - 5 + 3x^2 + 8x - 2$$ Combine like terms: $$4x^2 + 3x^2 = 7x^2$$ $$3x + 8x = 11x$$ $$-5 - 2 = -7$$ So, the simplified form is: $$7x^2 + 11x - 7$$ 2. Simplify the expression $(5x^2 - 9xy - 11y^2) - (7x^2 - 2xy + 2y^2)$. Distribute the minus sign: $$5x^2 - 9xy - 11y^2 - 7x^2 + 2xy - 2y^2$$ Combine like terms: $$5x^2 - 7x^2 = -2x^2$$ $$-9xy + 2xy = -7xy$$ $$-11y^2 - 2y^2 = -13y^2$$ Simplified expression: $$-2x^2 - 7xy - 13y^2$$ 3. Calculate the cost $C$ when $C = RT^2 - 3T + 8$, with $R=4$ and $T=-3$. Substitute values: $$C = 4(-3)^2 - 3(-3) + 8$$ Calculate powers and products: $$4 \times 9 + 9 + 8 = 36 + 9 + 8$$ Sum all terms: $$36 + 9 + 8 = 53$$ So, the cost is 53 dollars. 4. Daniel is $x$ years old. His brother is $x + 5$ years old. The product of their ages is given by: $$x(x + 5) = 71$$ Expand: $$x^2 + 5x = 71$$ Rewrite as a quadratic equation: $$x^2 + 5x - 71 = 0$$ Use the quadratic formula: $$x = \frac{-5 \pm \sqrt{5^2 - 4 \times 1 \times (-71)}}{2 \times 1} = \frac{-5 \pm \sqrt{25 + 284}}{2} = \frac{-5 \pm \sqrt{309}}{2}$$ Calculate approximate roots: $$\sqrt{309} \approx 17.58$$ Possible solutions: $$x = \frac{-5 + 17.58}{2} = 6.29$$ $$x = \frac{-5 - 17.58}{2} = -11.29$$ Since age cannot be negative, Daniel is approximately 6 years old. His brother's age: $$6 + 5 = 11$$ Final answers: 1) $7x^2 + 11x - 7$ 2) $-2x^2 - 7xy - 13y^2$ 3) 53 4) Daniel is 6 years old, his brother is 11 years old.