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Here's how to solve the problem: a) Find the values of the constants a and b. Step 1: Use the Remainder Theorem. When P(x) is divided by (x+3), the remainder is 18. This means P(-3) = 18. Substitute x = -3 into P(x) = 2x^3 + ax^2 + bx - 12: 2(-3)^3 + a(-3)^2 + b(-3) - 12 = 18 2(-27) + a(9) - 3b - 12 = 18 -54 + 9a - 3b - 12 = 18 9a - 3b - 66 = 18 9a - 3b = 84 Divide by 3: 3a - b = 28 (*) Step 2: Use the Factor Theorem. Given that (x+1) is a factor of P(x), this means P(-1) = 0. Substitute x = -1 into P(x) = 2x^3 + ax^2 + bx - 12: 2(-1)^3 + a(-1)^2 + b(-1) - 12 = 0 2(-1) + a(1) - b - 12 = 0 -2 + a - b - 12 = 0 a - b - 14 = 0 a - b = 14 (**) Step 3: Solve the system of linear equations. We have two equations: 1) 3a - b = 28 2) a - b = 14 Subtract equation (*) from equation (): (3a - b) - (a - b) = 28 - 14 2a = 14 a = 7 Substitute a = 7 into equation (**): 7 - b = 14 -b = 14 - 7 -b = 7 b = -7 The values of the constants are a=7, b=-7. b) Factorise P(x) completely. Step 4: Write the polynomial with the found constants. P(x) = 2x^3 + 7x^2 - 7x - 12 Step 5: Use synthetic division to divide P(x) by (x+1). Since (x+1) is a factor, we divide by x = -1. ` -1 | 2 7 -7 -12 | -2 -5 12 ------------------ 2 5 -12 0 ` The quotient is 2x^2 + 5x - 12. So, P(x) = (x+1)(2x^2 + 5x - 12). Step 6: Factorise the quadratic quotient 2x^2 + 5x - 12. We look for two numbers that multiply to 2 × (-12) = -24 and add up to 5. These numbers are 8 and -3. 2x^2 + 5x - 12 = 2x^2 + 8x - 3x - 12 = 2x(x+4) - 3(x+4) = (2x-3)(x+4) Step 7: Write the complete factorisation of P(x). P(x) = (x+1)(2x-3)(x+4) The complete factorisation is (x+1)(2x-3)(x+4). Send me the next one 📸