1.1) Simplify the following expresion fully: 1.1.1) \( (x+2)\left(x^{2}-x+3\right) \) (1.1.2) \( \frac{5}{x+3}-\frac{3}{2-x} \) 1.1.3) \( (2 y+3)\left(7 y^{2}-6 y-8\right) \) 1.1.14) \( \frac{x^{3}+1}{x^{2}+x+1} \) 1.1.5) \( \frac{3}{x^{2}-9}+\frac{2}{(x-3)^{2}} \)

Certainly! Let's simplify each of the given expressions step by step. --- ### **1.1.1) Simplify \( (x + 2)(x^{2} - x + 3) \)** **Steps:** 1. **Expand the expression using the distributive property (FOIL):** \[ (x + 2)(x^2 - x + 3) = x(x^2 - x + 3) + 2(x^2 - x + 3) \] 2. **Multiply each term:** \[ = x^3 - x^2 + 3x + 2x^2 - 2x + 6 \] 3. **Combine like terms:** \[ = x^3 + ( -x^2 + 2x^2 ) + (3x - 2x) + 6 = x^3 + x^2 + x + 6 \] **Simplified Expression:** \[ x^3 + x^2 + x + 6 \] --- ### **1.1.2) Simplify \( \frac{5}{x + 3} - \frac{3}{2 - x} \)** **Steps:** 1. **Rewrite the second denominator to match the form \((x - 2)\):** \[ 2 - x = -(x - 2) \] Thus, \[ \frac{5}{x + 3} - \frac{3}{2 - x} = \frac{5}{x + 3} + \frac{3}{x - 2} \] 2. **Find a common denominator:** \[ \text{Common Denominator} = (x + 3)(x - 2) \] 3. **Combine the fractions:** \[ = \frac{5(x - 2) + 3(x + 3)}{(x + 3)(x - 2)} = \frac{5x - 10 + 3x + 9}{(x + 3)(x - 2)} = \frac{8x - 1}{(x + 3)(x - 2)} \] **Simplified Expression:** \[ \frac{8x - 1}{(x + 3)(x - 2)} \] --- ### **1.1.3) Simplify \( (2y + 3)(7y^{2} - 6y - 8) \)** **Steps:** 1. **Expand the expression using the distributive property:** \[ (2y + 3)(7y^2 - 6y - 8) = 2y(7y^2 - 6y - 8) + 3(7y^2 - 6y - 8) \] 2. **Multiply each term:** \[ = 14y^3 - 12y^2 - 16y + 21y^2 - 18y - 24 \] 3. **Combine like terms:** \[ = 14y^3 + ( -12y^2 + 21y^2 ) + ( -16y - 18y ) - 24 = 14y^3 + 9y^2 - 34y - 24 \] **Simplified Expression:** \[ 14y^3 + 9y^2 - 34y - 24 \] --- ### **1.1.14) Simplify \( \frac{x^{3} + 1}{x^{2} + x + 1} \)** **Steps:** 1. **Factor the numerator using the sum of cubes formula:** \[ x^3 + 1 = (x + 1)(x^2 - x + 1) \] 2. **Rewrite the expression:** \[ \frac{(x + 1)(x^2 - x + 1)}{x^2 + x + 1} \] 3. **Notice that \(x^2 - x + 1\) and \(x^2 + x + 1\) do not cancel out directly. Perform polynomial division or recognize that:** \[ x^3 + 1 = (x - 1)(x^2 + x + 1) + 2 \] Therefore, \[ \frac{x^3 + 1}{x^2 + x + 1} = x - 1 + \frac{2}{x^2 + x + 1} \] **Simplified Expression:** \[ x - 1 + \frac{2}{x^2 + x + 1} \] --- ### **1.1.5) Simplify \( \frac{3}{x^{2} - 9} + \frac{2}{(x - 3)^{2}} \)** **Steps:** 1. **Factor the denominator \(x^2 - 9\):** \[ x^2 - 9 = (x - 3)(x + 3) \] 2. **Rewrite the expression:** \[ \frac{3}{(x - 3)(x + 3)} + \frac{2}{(x - 3)^2} \] 3. **Find a common denominator, which is \((x - 3)^2(x + 3)\):** \[ = \frac{3(x - 3)}{(x - 3)^2(x + 3)} + \frac{2(x + 3)}{(x - 3)^2(x + 3)} \] 4. **Combine the fractions:** \[ = \frac{3(x - 3) + 2(x + 3)}{(x - 3)^2(x + 3)} = \frac{3x - 9 + 2x + 6}{(x - 3)^2(x + 3)} = \frac{5x - 3}{(x - 3)^2(x + 3)} \] **Simplified Expression:** \[ \frac{5x - 3}{(x + 3)(x - 3)^2} \] --- Each expression has been fully simplified as shown above.