Question

Perform the following operations and prove closure. Show your work. (x/x+3) + (x+2/x+5)

Answer 1

Answer:

$\frac{2(x^{2} + 5x + 3)}{(x+3)(x+5)}$

Step-by-step explanation:

We need to sum the following two expressions:

$\frac{x}{x+3} + \frac{x+2}{x+5}$

$\frac{x(x+5) + (x+2)(x+3)}{(x+3)(x+5)}$

expanding the polynomial in the numerator:

$\frac{2x^{2} + 10x + 6}{(x+3)(x+5)}$

$\frac{2(x^{2} + 5x + 3)}{(x+3)(x+5)}$

This is the most simplified form we can get:

$\frac{2(x^{2} + 5x + 3)}{(x+3)(x+5)}$

Answer 2

Answer:

$\frac{2(x^{2}+5x+3)}{(x+3)(x+5)}$

Step-by-step explanation:

The given expression is $\frac{x}{x+3}+\frac{x+2}{x+5}$

We have to simplify the given expression

$\frac{x}{x+3}+\frac{x+2}{x+5}$

= $\frac{x(x+5)+(x+2)(x+3)}{(x+3)(x+5)}$         [Distributive law]

= $\frac{x^{2}+5x+x^{2}+3x+2x+6}{(x+3)(x+5)}$

= $\frac{2x^{2}+10x+6}{(x+3)(x+5)}$

= $\frac{2(x^{2}+5x+3)}{(x+3)(x+5)}$

Finally the simplified form of the given expression is $\frac{2(x^{2}+5x+3)}{(x+3)(x+5)}$