Expand and simplify the expressions: (x-y)2, (4x-2y)2. Solve for x: x2+5x+6=0, 6/(x-4) = (x+4)/x

Here are the solutions for the given expressions and equations: 1. Expand and simplify the following expressions: a) (x-y)^2 Step 1: Use the binomial expansion formula (a-b)^2 = a^2 - 2ab + b^2. Here, a=x and b=y. (x-y)^2 = x^2 - 2(x)(y) + y^2 Step 2: Simplify the expression. x^2 - 2xy + y^2 The expanded and simplified expression is x^2 - 2xy + y^2. b) (4x-2y)^2 Step 1: Use the binomial expansion formula (a-b)^2 = a^2 - 2ab + b^2. Here, a=4x and b=2y. (4x-2y)^2 = (4x)^2 - 2(4x)(2y) + (2y)^2 Step 2: Simplify each term. (4x)^2 = 16x^2 2(4x)(2y) = 16xy (2y)^2 = 4y^2 Step 3: Combine the simplified terms. 16x^2 - 16xy + 4y^2 The expanded and simplified expression is 16x^2 - 16xy + 4y^2. 2. Solve for x: a) x^2+5x+6=0 Step 1: Factor the quadratic expression. Find two numbers that multiply to 6 and add to 5. These numbers are 2 and 3. (x+2)(x+3) = 0 Step 2: Set each factor equal to zero and solve for x. x+2 = 0 or x+3 = 0 x = -2 or x = -3 The solutions are x = -2 or x = -3. b) (6)/(x-4) = (x+4)/(x) Step 1: Cross-multiply to eliminate the denominators. Note that x ≠ 4 and x ≠ 0. 6x = (x+4)(x-4) Step 2: Expand the right side of the equation. 6x = x^2 - 4x + 4x - 16 6x = x^2 - 16 Step 3: Rearrange the equation into standard quadratic form ax^2+bx+c=0. x^2 - 6x - 16 = 0 Step 4: Factor the quadratic expression. Find two numbers that multiply to -16 and add to -6. These numbers are -8 and 2. (x-8)(x+2) = 0 Step 5: Set each factor equal to zero and solve for x. x-8 = 0 or x+2 = 0 x = 8 or x = -2 Both solutions are valid as they do not make the original denominators zero. The solutions are x = 8 or x = -2. 3 done, 2 left today. You're making progress.