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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You're on a roll — here are the factorisations and simplifications: Factorise completely. 3.1 x^2 - x - 12 Step 1: Find two numbers that multiply to -12 and add to -1. These numbers are 3 and -4. x^2 - x - 12 = (x + 3)(x - 4) The factorised expression is (x + 3)(x - 4). 3.2 2p^2 - 60p - 128 Step 1: Factor out the greatest common factor (GCF), which is 2. 2p^2 - 60p - 128 = 2(p^2 - 30p - 64) Step 2: Factor the quadratic expression p^2 - 30p - 64. Find two numbers that multiply to -64 and add to -30. These numbers are 2 and -32. = 2(p + 2)(p - 32) The factorised expression is 2(p + 2)(p - 32). 3.2 (x - 1)^2 - 2(x - 1) - 8 Step 1: Let y = (x - 1). Substitute y into the expression. y^2 - 2y - 8 Step 2: Factor the quadratic expression. Find two numbers that multiply to -8 and add to -2. These numbers are 2 and -4. (y + 2)(y - 4) Step 3: Substitute back y = (x - 1). ((x - 1) + 2)((x - 1) - 4) = (x + 1)(x - 5) The factorised expression is (x + 1)(x - 5). QUESTION 4: Simplify completely. 4.1 (4x^2 + 8x)/(2x) Step 1: Factor the numerator by taking out the common factor 4x. (4x(x + 2))/(2x) Step 2: Cancel out the common term 2x from the numerator and the denominator. = (2(x + 2))/(1) = 2(x + 2) The simplified expression is 2(x + 2). 4.2 (x^2 - 2x - 15)/(x^2 - 25) Step 1: Factor the numerator x^2 - 2x - 15. Find two numbers that multiply to -15 and add to -2. These are 3 and -5. x^2 - 2x - 15 = (x + 3)(x - 5) Step 2: Factor the denominator x^2 - 25. This is a difference of squares, a^2 - b^2 = (a - b)(a + b). x^2 - 25 = (x - 5)(x + 5) Step 3: Substitute the factored expressions back into the fraction and simplify. ((x + 3)(x - 5))/((x - 5)(x + 5)) = (x + 3)/(x + 5) The simplified expression is (x + 3)/(x + 5). 4.3 (x(x-1) - (1-x))/(5x^2 - 5) Step 1: Simplify the numerator. Note that (1 - x) = -(x - 1). x(x - 1) - (-(x - 1)) = x(x - 1) + (x - 1) Factor out the common term (x - 1). = (x - 1)(x + 1) Step 2: Factor the denominator 5x^2 - 5. Factor out the GCF 5, then recognize the difference of squares. 5x^2 - 5 = 5(x^2 - 1) = 5(x - 1)(x + 1) Step 3: Substitute the factored expressions back into the fraction and simplify. ((x - 1)(x + 1))/(5(x - 1)(x + 1)) = (1)/(5) The simplified expression is (1)/(5). 4.4 (x^2 + 8x + 15)/(3x^2 + 9x) + (x^2 - 25)/(6x^2 + 42x) Step 1: Simplify the first fraction (x^2 + 8x + 15)/(3x^2 + 9x). Factor the numerator: x^2 + 8x + 15 = (x + 3)(x + 5). Factor the denominator: 3x^2 + 9x = 3x(x + 3). So, the first fraction becomes: ((x + 3)(x + 5))/(3x(x + 3)) = (x + 5)/(3x). Step 2: Simplify the second fraction (x^2 - 25)/(6x^2 + 42x). Factor the numerator: x^2 - 25 = (x - 5)(x + 5). Factor the denominator: 6x^2 + 42x = 6x(x + 7). So, the second fraction becomes: ((x - 5)(x + 5))/(6x(x + 7)). Step 3: Add the simplified fractions: (x + 5)/(3x) + ((x - 5)(x + 5))/(6x(x + 7)). Find the least common denominator (LCD), which is 6x(x + 7). Rewrite the first fraction with the LCD: ((x + 5))/(3x) · (2(x + 7))/(2(x + 7)) = (2(x + 5)(x + 7))/(6x(x + 7)) Step 4: Combine the fractions over the common denominator. (2(x + 5)(x + 7) + (x - 5)(x + 5))/(6x(x + 7)) Step 5: Factor out (x + 5) from the numerator. ((x + 5)[2(x + 7) + (x - 5)])/(6x(x + 7)) Step 6: Simplify the expression inside the square brackets. 2(x + 7) + (x - 5) = 2x + 14 + x - 5 = 3x + 9 Factor out 3 from 3x + 9: 3(x + 3). Step 7: Substitute this back into the numerator and simplify the entire expression. ((x + 5) · 3(x + 3))/(6x(x + 7)) = (3(x + 3)(x + 5))/(6x(x + 7)) = ((x + 3)(x + 5))/(2x(x + 7)) The simplified expression is $(x