1. **Expand and Simplify:** a) Expand and simplify $ (3x^2 - 4xy + 5y^2) - (5x^2 - 7xy - 4y^2) $. Use distributive property to remove parentheses: $$3x^2 - 4xy + 5y^2 - 5x^2 + 7xy + 4y^2$$ Combine like terms: $$ (3x^2 - 5x^2) + (-4xy + 7xy) + (5y^2 + 4y^2) = -2x^2 + 3xy + 9y^2$$ b) Expand $ -3x^2 y (5x^3 + 3y^2) $. Multiply each term inside the parentheses by $ -3x^2 y $: $$ -3x^2 y \times 5x^3 = -15x^{5} y $$ $$ -3x^2 y \times 3y^2 = -9x^2 y^{3} $$ So the expression is: $$ -15x^{5} y - 9x^2 y^{3} $$ c) Expand $ -2(x - 2y)(x + 5y) $. First expand $ (x - 2y)(x + 5y) $ using FOIL: $$ x \times x = x^2 $$ $$ x \times 5y = 5xy $$ $$ -2y \times x = -2xy $$ $$ -2y \times 5y = -10y^2 $$ Combine like terms inside parentheses: $$ x^2 + (5xy - 2xy) - 10y^2 = x^2 + 3xy - 10y^2 $$ Now multiply by $ -2 $: $$ -2(x^2 + 3xy - 10y^2) = -2x^2 - 6xy + 20y^2 $$ 2. **Factor each expression:** a) $ 6x^2 - 3x $ Factor out the greatest common factor (GCF) $3x$: $$ 3x(2x - 1) $$ b) $ x^2 + 7x - 18 $ Find two numbers that multiply to $-18$ and add to $7$: $9$ and $-2$. Factor: $$ (x + 9)(x - 2) $$ c) $ 49x^2 - 81y^2 $ Recognize difference of squares: $$ (7x)^2 - (9y)^2 = (7x - 9y)(7x + 9y) $$ d) $ 2x^2 + 5x + 2 $ Find two numbers that multiply to $2 \times 2 = 4$ and add to $5$: $4$ and $1$. Rewrite middle term: $$ 2x^2 + 4x + x + 2 $$ Factor by grouping: $$ 2x(x + 2) + 1(x + 2) = (2x + 1)(x + 2) $$ 3. **Roots of quadratic equation:** Given $ y = 2(x - 5)(x + 4) $, roots are values of $x$ that make $y=0$. Set each factor to zero: $$ x - 5 = 0 \Rightarrow x = 5 $$ $$ x + 4 = 0 \Rightarrow x = -4 $$ So roots are $5$ and $-4$. 4. **Axis of symmetry:** Given roots $-6$ and $16$, axis of symmetry is the vertical line halfway between roots: $$ x = \frac{-6 + 16}{2} = \frac{10}{2} = 5 $$ 5. **Discriminant:** For quadratic $ x^2 + 11x + 24 = 0 $, discriminant $\Delta$ is: $$ \Delta = b^2 - 4ac $$ Here, $a=1$, $b=11$, $c=24$. Calculate: $$ \Delta = 11^2 - 4(1)(24) = 121 - 96 = 25 $$ 6. **Parabola $ y = -3(x - 2)^2 + 5 $:** a) Direction of opening: Since coefficient of squared term is $-3$ (negative), parabola opens downward. b) Vertex coordinates: Vertex form is $ y = a(x - h)^2 + k $, vertex at $(h, k)$. Here, $h=2$, $k=5$, so vertex is $(2, 5)$. c) Axis of symmetry: Vertical line through vertex $x = 2$. d) Maximum or minimum: Since parabola opens downward, vertex is a maximum point. e) Maximum value: The $y$-coordinate of vertex is maximum value, $5$. 7. **Function evaluations:** Given $ f(x) = x^2 + 3x + 2 $ and $ g(x) = 4x - 3 $. a) Calculate $ f(2) $: $$ f(2) = 2^2 + 3(2) + 2 = 4 + 6 + 2 = 12 $$ b) Calculate $ g(-5) $: $$ g(-5) = 4(-5) - 3 = -20 - 3 = -23 $$