This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here are the solutions to the problems. Question 1: Substitution 1.1 Step 1: Substitute x=3 into the expression 2(x+4)^2. 2(3+4)^2 Step 2: Perform the operation inside the parenthesis. 2(7)^2 Step 3: Calculate the square. 2(49) Step 4: Multiply. 98 The value is: 98 1.2 Step 1: Substitute x=8 and y=9 into the expression [3]x+7sqrt(y). [3]8+7sqrt(9) Step 2: Simplify the terms under the radicals. [3]15sqrt(9) Step 3: Calculate the cube root and the square root. [3]153 The value is: [3]153 1.3 Step 1: Substitute x=2 and y=3 into the expression 6x^4y^2 - 18x^3y^3 + 12x^2y. 6(2)^4(3)^2 - 18(2)^3(3)^3 + 12(2)^2(3) Step 2: Calculate the powers. 6(16)(9) - 18(8)(27) + 12(4)(3) Step 3: Perform the multiplications in each term. 864 - 3888 + 144 Step 4: Perform the additions and subtractions. -3024 + 144 -2880 The value is: -2880 Question 2: Expanding Brackets 2.1 Step 1: Expand the product (x+4)(x-3) using the FOIL method (First, Outer, Inner, Last). x · x + x · (-3) + 4 · x + 4 · (-3) x^2 - 3x + 4x - 12 Step 2: Combine like terms. x^2 + x - 12 The expanded expression is: x^2+x-12 2.2 Step 1: Distribute x^2 to each term inside the parenthesis (2x-4). x^2(2x) - x^2(4) 2x^3 - 4x^2 The expanded expression is: 2x^3-4x^2 Question 3: Factorize the following expressions 3.1 Step 1: Find the greatest common factor (GCF) of x^2 and 4x. The GCF is x. Step 2: Factor out the GCF. x(x+4) The factored expression is: x(x+4) 3.2 Step 1: Find the GCF of 25xy^2 and 15x^2y. The GCF of 25 and 15 is 5. The GCF of x and x^2 is x. The GCF of y^2 and y is y. So, the overall GCF is 5xy. Step 2: Factor out the GCF. 5xy(5y + 3x) The factored expression is: 5xy(5y+3x) 3.3 Step 1: Find the GCF of 8x^3y^2 and 12x^2y^3. The GCF of 8 and 12 is 4. The GCF of x^3 and x^2 is x^2. The GCF of y^2 and y^3 is y^2. So, the overall GCF is 4x^2y^2. Step 2: Factor out the GCF. 4x^2y^2(2x - 3y) The factored expression is: 4x^2y^2(2x-3y) 3.4 Step 1: Factor the quadratic expression x^2-x-12. We need two numbers that multiply to -12 and add to -1. These numbers are 3 and -4. (x+3)(x-4) The factored expression is: (x+3)(x-4) 3.5 Step 1: Find the GCF of 6x^4y^2, -18x^3y^3, and 12x^2y. The GCF of 6, 18, and 12 is 6. The GCF of x^4, x^3, and x^2 is x^2. The GCF of y^2, y^3, and y is y. So, the overall GCF is 6x^2y. Step 2: Factor out the GCF. 6x^2y(x^2y - 3xy^2 + 2) The factored expression is: 6x^2y(x^2y-3xy^2+2) 3.6 Step 1: Factor the quadratic expression x^2-9x+20. We need two numbers that multiply to 20 and add to -9. These numbers are -4 and -5. (x-4)(x-5) The factored expression is: (x-4)(x-5) 3.7 Step 1: Subtract x^2-4x-5 from 2x^2+x-1. (2x^2+x-1) - (x^2-4x-5) Step 2: Distribute the negative sign. 2x^2+x-1 - x^2+4x+5 Step 3: Combine like terms. (2x^2-x^2) + (x+4x) + (-1+5) x^2 + 5x + 4 Step 4: Factor the resulting quadratic expression x^2+5x+4. We need two numbers that multiply to 4 and add to 5. These numbers are 1 and 4. (x+1)(x+4) The factored expression is: (x+1)(x+4) That's 2 down. 3 left today — send the next one.