1. Expand and Simplify each expression: 1.a) Expand and simplify $ (3x^2 - 4xy + 5y^2) - (5x^2 - 7xy - 4y^2) $. Step 1: Remove parentheses carefully, changing signs for the second group: $$3x^2 - 4xy + 5y^2 - 5x^2 + 7xy + 4y^2$$ Step 2: Combine like terms: $$ (3x^2 - 5x^2) + (-4xy + 7xy) + (5y^2 + 4y^2) = -2x^2 + 3xy + 9y^2$$ Answer: $-2x^2 + 3xy + 9y^2$ 1.b) Simplify $-3x^2 y(5x^3 + 3y^2)$. Step 1: Distribute $-3x^2 y$ to each term inside the parentheses: $$-3x^2 y \times 5x^3 = -15x^{5} y$$ $$-3x^2 y \times 3y^2 = -9x^2 y^{3}$$ Answer: $-15x^{5} y - 9x^{2} y^{3}$ 1.c) Simplify $-2(x - 2y)(x + 5y)$. Step 1: Expand the binomials: $$(x - 2y)(x + 5y) = x^2 + 5xy - 2xy - 10y^2 = x^2 + 3xy - 10y^2$$ Step 2: Multiply by $-2$: $$-2(x^2 + 3xy - 10y^2) = -2x^2 - 6xy + 20y^2$$ Answer: $-2x^2 - 6xy + 20y^2$ 2. Factor each expression: 2.a) Factor $6x^2 - 3x$. Step 1: Find common factor $3x$: $$3x(2x - 1)$$ Answer: $3x(2x - 1)$ 2.b) Factor $x^2 + 7x - 18$. Step 1: Find two numbers that multiply to $-18$ and add to $7$: $9$ and $-2$. Step 2: Factor: $$(x + 9)(x - 2)$$ Answer: $(x + 9)(x - 2)$ 2.c) Factor $49x^2 - 81y^2$. Step 1: Recognize difference of squares: $$ (7x)^2 - (9y)^2 = (7x - 9y)(7x + 9y)$$ Answer: $(7x - 9y)(7x + 9y)$ 2.d) Factor $2x^2 + 5x + 2$. Step 1: Find two numbers that multiply to $2 imes 2 = 4$ and add to $5$: $4$ and $1$. Step 2: Rewrite middle term: $$2x^2 + 4x + x + 2$$ Step 3: Factor by grouping: $$2x(x + 2) + 1(x + 2) = (2x + 1)(x + 2)$$ Answer: $(2x + 1)(x + 2)$ 3. Find roots of $y = 2(x - 5)(x + 4)$. Step 1: Set $y=0$ to find roots: $$2(x - 5)(x + 4) = 0 \\ (x - 5)(x + 4) = 0$$ Step 2: Solve each factor: $$x - 5 = 0 \Rightarrow x = 5$$ $$x + 4 = 0 \Rightarrow x = -4$$ Answer: $5$ and $-4$ 4. Equation of axis of symmetry for roots $-6$ and $16$. Step 1: Axis of symmetry is the vertical line halfway between roots: $$x = \frac{-6 + 16}{2} = \frac{10}{2} = 5$$ Answer: $x = 5$ 5. Calculate discriminant of $x^2 + 11x + 24 = 0$. Step 1: Recall discriminant formula: $$\Delta = b^2 - 4ac$$ Step 2: Identify coefficients: $a=1$, $b=11$, $c=24$. Step 3: Calculate: $$\Delta = 11^2 - 4(1)(24) = 121 - 96 = 25$$ Answer: $25$ 6. For parabola $y = -3(x - 2)^2 + 5$: 6.a) Direction of opening: Coefficient of squared term is $-3$ (negative), so parabola opens downward. Answer: Opens downward 6.b) Coordinates of vertex: Vertex form is $y = a(x - h)^2 + k$, vertex at $(h, k)$. Here, $h=2$, $k=5$. Answer: $(2, 5)$ 6.c) Equation of axis of symmetry: Vertical line through vertex $x = h = 2$. Answer: $x = 2$ 6.d) Maximum or minimum value: Since parabola opens downward, vertex is a maximum point. Answer: Maximum 6.e) Maximum or minimum value: Value of $y$ at vertex is $5$. Answer: $5$ 7. Given $f(x) = x^2 + 3x + 2$ and $g(x) = 4x - 3$: 7.a) Find $f(2)$: $$f(2) = 2^2 + 3(2) + 2 = 4 + 6 + 2 = 12$$ Answer: $12$ 7.b) Find $g(-5)$: $$g(-5) = 4(-5) - 3 = -20 - 3 = -23$$ Answer: $-23$