1.5 Simplify the following expressions fully; 1.5.1 \( (2 x-1)\left(x^{2}-2 x+1\right) \) \( 1.5 .2(3 x+5)^{2} \) \( 1.5 .3 \quad \frac{2^{x}-2^{x-2}}{2^{x+1}-2^{x}} \) \( 1.5 .4 \frac{3}{a-4}+\frac{2}{a+3}-\frac{21}{a^{2}-a-12} \)

Sure! Let’s simplify each expression step by step. **1.5.1** \( (2x - 1)(x^{2} - 2x + 1) \) First, recognize that \( x^{2} - 2x + 1 \) can be factored as \( (x - 1)^{2} \). Now we have: \[ (2x - 1)(x - 1)^{2} = (2x - 1)(x - 1)(x - 1) \] Expanding this gives: \[ = (2x - 1)(x^{2} - 2x + 1) = 2x(x^{2} - 2x + 1) - (x^{2} - 2x + 1) = 2x^{3} - 4x^{2} + 2x - x^{2} + 2x - 1 = 2x^{3} - 5x^{2} + 4x - 1 \] So, the simplified result is: \[ 2x^{3} - 5x^{2} + 4x - 1 \] **1.5.2** \( (3x + 5)^{2} \) This expression can be simplified using the square of a binomial formula: \[ (a + b)^{2} = a^{2} + 2ab + b^{2} \] Applying it here: \[ (3x + 5)^{2} = (3x)^{2} + 2(3x)(5) + (5)^{2} = 9x^{2} + 30x + 25 \] Thus, the result is: \[ 9x^{2} + 30x + 25 \] **1.5.3** \( \frac{2^{x}-2^{x-2}}{2^{x+1}-2^{x}} \) First, factor the numerator: \[ = \frac{2^{x}(1 - 2^{-2})}{2^{x}(2 - 1)} = \frac{2^{x}(1 - \frac{1}{4})}{2^{x}} = \frac{2^{x} \cdot \frac{3}{4}}{2^{x}} = \frac{3}{4} \] So, the simplified result is: \[ \frac{3}{4} \] **1.5.4** \( \frac{3}{a-4} + \frac{2}{a+3} - \frac{21}{a^{2}-a-12} \) First, factor the denominator of the last term: \[ a^{2} - a - 12 = (a - 4)(a + 3) \] Now, rewriting the expression: \[ = \frac{3}{a-4} + \frac{2}{a+3} - \frac{21}{(a-4)(a+3)} \] To combine these fractions, find a common denominator, which is \( (a-4)(a+3) \): \[ = \frac{3(a + 3) + 2(a - 4) - 21}{(a-4)(a+3)} = \frac{3a + 9 + 2a - 8 - 21}{(a-4)(a+3)} = \frac{5a - 20}{(a-4)(a+3)} = \frac{5(a - 4)}{(a-4)(a+3)} = \frac{5}{a + 3}, \; a \neq 4 \] Thus, the simplified result is: \[ \frac{5}{a + 3}, \; a \neq 4 \]