Factor the numerator by taking out the common factor of -3.

Here are the factorizations and simplifications for each expression: a) To factorize and simplify (-3a + 18)/(a^2 - 9a + 18): Step 1: Factor the numerator by taking out the common factor of -3. -3a + 18 = -3(a - 6) Step 2: Factor the denominator. We need two numbers that multiply to 18 and add to -9. These numbers are -3 and -6. a^2 - 9a + 18 = (a - 3)(a - 6) Step 3: Substitute the factored expressions back into the fraction and simplify by canceling the common factor (a-6). (-3(a - 6))/((a - 3)(a - 6)) = (-3)/(a - 3) The simplified expression is (-3)/(a - 3). b) To factorize and simplify (4x^2 - 1)/(4x^2 - 8x): Step 1: Factor the numerator. 4x^2 - 1 is a difference of squares, (2x)^2 - 1^2. 4x^2 - 1 = (2x - 1)(2x + 1) Step 2: Factor the denominator by taking out the common factor of 4x. 4x^2 - 8x = 4x(x - 2) Step 3: Substitute the factored expressions back into the fraction. ((2x - 1)(2x + 1))/(4x(x - 2)) There are no common factors to cancel. The simplified expression is ((2x - 1)(2x + 1))/(4x(x - 2)). c) To factorize and simplify (a^2 - 25)/((a-5)^2) × (a-5)/(5+a): Step 1: Factor a^2 - 25 in the first numerator. It is a difference of squares. a^2 - 25 = (a - 5)(a + 5) Step 2: Rewrite the expression with the factored term. Note that 5+a is the same as a+5. ((a - 5)(a + 5))/((a-5)^2) × (a-5)/(a+5) Step 3: Simplify by canceling common factors. ((a - 5)(a + 5))/((a-5)(a-5)) × (a-5)/(a+5) = 1 The simplified expression is 1. d) To factorize and simplify (x^2 - 1)/((x-1)^2): Step 1: Factor the numerator. x^2 - 1 is a difference of squares. x^2 - 1 = (x - 1)(x + 1) Step 2: Rewrite the expression with the factored term. ((x - 1)(x + 1))/((x-1)^2) Step 3: Simplify by canceling one common factor of (x-1) from the numerator and denominator. (x + 1)/(x - 1) The simplified expression is (x + 1)/(x - 1). That's 2 down. 3 left today — send the next one.