Solve the equation: \$(x^2 + 3x)/(x^2 - 4) - (2x)/(x + 2) = 0\$.

The given equation

\$(x^2 + 3x)/(x^2 - 4) - (2x)/(x + 2) = 0\$

To solve this equation, let’s find the least common denominator (LCD) of the rational expressions, which is \$(x^2 - 4)(x + 2)\$:

\$ ((x^2 + 3x)(x + 2) - 2x(x^2 - 4)) / ((x^2 - 4)(x + 2)) = 0 \$

Expanding and simplifying the numerator:

\$ (x^3 + 2x^2 + 3x^2 + 6x - 2x^3 + 8x) / ((x^2 - 4)(x + 2)) = 0 \$

\$ (- x^3 + 5x^2 +14x) / ((x^2 - 4)(x + 2)) = 0 \$

After moving the denominator to the right side, and then simplified

\$ - x^3 + 5x^2 +14x = 0 \$

\$ x (- x^2 + 5x +14) = 0 \$

\$ x = 0 , - x^2 + 5x +14 = 0 \$

Now find the x value from the second factor, then taking (x - 7) is common in the equation

\$- x^2 + 5x +14 = 0 \$

\$ x^2 - 5x -14 = 0 \$

\$ (x - 7)(x + 2) = 0 \$

Setting each factor equal to zero gives us two possible solutions:

\$ x - 7 = 0 , x + 2 = 0 \$

\$ x = 7 , x = - 2 \$

Therefore, the solutions to the equation \$(x^2 + 3x)/(x^2 - 4) - (2x)/(x + 2) = 0\$ are \$x = 0, x = 7, x =- 2\$.

In our solution, we found x = 0 but not x = − 2 or
x = 7, So option A is incorrect

0, - 4, 5

In our solution, we found x = 0, x = − 2, and x = 7, but not x = − 5 or x = − 1, So option B is incorrect

0, - 5, -1

This option matches the solutions we found: x = 0, x = 7, and x = − 2, So option C is correct

0, 7, - 2

This is not accurate as it misrepresents the solutions found for the given equation

0, 2, - 7

Option

C

Can you summarize what you’ve understood in the above steps?