1. **State the problem:** Simplify the algebraic expression $$\frac{3x^2}{x+4} + \frac{5z}{x+64}$$ and $$\frac{2x-3}{x^2+5x+6} + \frac{4}{x+3}$$ and $$\frac{6}{x+4} + \frac{2x+8}{x+4} - \frac{6x+2}{x+4}$$ and $$\frac{x+4}{x-5} + \frac{x+5}{x+3} + 3 + \frac{x+8}{x^2-25}$$. 2. **Important rules:** - To add or subtract algebraic fractions, find a common denominator. - Factor denominators when possible to simplify. - Use the property $$\frac{a}{c} + \frac{b}{c} = \frac{a+b}{c}$$ when denominators are the same. 3. **Simplify each expression step-by-step:** **Expression 1:** $$\frac{3x^2}{x+4} + \frac{5z}{x+64}$$ - Denominators are different and cannot be factored to a common denominator easily. - So, the expression remains as is or combined over the product denominator: $$\frac{3x^2(x+64)}{(x+4)(x+64)} + \frac{5z(x+4)}{(x+64)(x+4)} = \frac{3x^2(x+64) + 5z(x+4)}{(x+4)(x+64)}$$ **Expression 2:** $$\frac{2x-3}{x^2+5x+6} + \frac{4}{x+3}$$ - Factor denominator: $$x^2+5x+6 = (x+2)(x+3)$$ - Rewrite first fraction: $$\frac{2x-3}{(x+2)(x+3)}$$ - Second fraction denominator is $$x+3$$, multiply numerator and denominator by $$x+2$$ to get common denominator: $$\frac{4(x+2)}{(x+3)(x+2)}$$ - Add numerators: $$\frac{2x-3 + 4(x+2)}{(x+2)(x+3)} = \frac{2x-3 + 4x + 8}{(x+2)(x+3)} = \frac{6x + 5}{(x+2)(x+3)}$$ **Expression 3:** $$\frac{6}{x+4} + \frac{2x+8}{x+4} - \frac{6x+2}{x+4}$$ - All denominators are the same, so combine numerators: $$\frac{6 + (2x+8) - (6x+2)}{x+4} = \frac{6 + 2x + 8 - 6x - 2}{x+4} = \frac{(6 + 8 - 2) + (2x - 6x)}{x+4} = \frac{12 - 4x}{x+4}$$ - Factor numerator: $$\frac{4(3 - x)}{x+4}$$ **Expression 4:** $$\frac{x+4}{x-5} + \frac{x+5}{x+3} + 3 + \frac{x+8}{x^2-25}$$ - Factor denominator $$x^2-25 = (x-5)(x+3)$$ - Rewrite $$3$$ as $$\frac{3(x-5)(x+3)}{(x-5)(x+3)}$$ - Find common denominator $$ (x-5)(x+3) $$ - Rewrite first two fractions: $$\frac{(x+4)(x+3)}{(x-5)(x+3)} + \frac{(x+5)(x-5)}{(x+3)(x-5)}$$ - Add all fractions: $$\frac{(x+4)(x+3) + (x+5)(x-5) + 3(x-5)(x+3) + (x+8)}{(x-5)(x+3)}$$ - Expand numerators: $$(x+4)(x+3) = x^2 + 7x + 12$$ $$(x+5)(x-5) = x^2 - 25$$ $$3(x-5)(x+3) = 3(x^2 - 2x - 15) = 3x^2 - 6x - 45$$ - Sum numerator: $$x^2 + 7x + 12 + x^2 - 25 + 3x^2 - 6x - 45 + x + 8 = (x^2 + x^2 + 3x^2) + (7x - 6x + x) + (12 - 25 - 45 + 8) = 5x^2 + 2x - 50$$ - Final expression: $$\frac{5x^2 + 2x - 50}{(x-5)(x+3)}$$ - Factor numerator: $$5x^2 + 2x - 50 = 5(x^2 + \frac{2}{5}x - 10)$$ (cannot factor nicely further) 4. **Final answers:** - Expression 1: $$\frac{3x^2(x+64) + 5z(x+4)}{(x+4)(x+64)}$$ - Expression 2: $$\frac{6x + 5}{(x+2)(x+3)}$$ - Expression 3: $$\frac{4(3 - x)}{x+4}$$ - Expression 4: $$\frac{5x^2 + 2x - 50}{(x-5)(x+3)}$$