Solve the Equation: 3/(x+1)^2 + 2x/(x+1) + x + 1 = 3 Step-by-Step

To solve the equation $\frac{3}{(x+1)^2}+\frac{2x}{x+1}+x+1=3$, we will clear the fractions by finding a common denominator.

• Step 1: The common denominator of the fractions $\frac{3}{(x+1)^2}$ and $\frac{2x}{x+1}$ is $(x+1)^2$.
• Step 2: Multiply each term in the equation by $(x+1)^2$ to clear the fractions:
  $\left(\frac{3}{(x+1)^2} \cdot (x+1)^2\right) + \left(\frac{2x}{x+1} \cdot (x+1)^2\right) + (x+1) \cdot (x+1)^2 = 3 \cdot (x+1)^2$.
• Step 3: Simplify each term:
  - The first term becomes $3$.
  - The second term becomes $2x(x+1)$.
  - The third term becomes $(x+1)^3$.
  Then, equate to the right-hand side: $3 + 2x(x+1) + (x+1)^3 = 3(x+1)^2$.
• Step 4: Expand the expressions:
  - Expand $2x(x+1)$ to get $2x^2 + 2x$.
  - Expand $(x+1)^3$ to $x^3 + 3x^2 + 3x + 1$.
  - Expand $3(x+1)^2$ to $3(x^2 + 2x + 1)$ or $3x^2 + 6x + 3$.
• Step 5: Formulate the new equation by bringing all terms to one side:
  $x^3 + 3x^2 + 3x + 1 + 2x^2 + 2x + 3 - 3x^2 - 6x - 3 = 0$.
• Combine like terms to simplify:
  $x^3 + 2x^2 - x + 1 = 0$
• Step 6: Solve the resulting equation, which is already simplified:
  Factor the equation if possible. Here we substitute likely values or use a factoring method.
  Using the Rational Root Theorem or graphically analyzing roots might give viable real solutions.
  Let's factor even further:
  $(x - (\sqrt{3}-2))(x - (-\sqrt{3}-2)) = 0$.
• Step 7: Solve for $x$ from the factors:
  $x = \sqrt{3} - 2$ or $x = -\sqrt{3} - 2$.

Thus, the values of $x$ that satisfy this equation are $x = \sqrt{3} - 2$ and $x = -\sqrt{3} - 2$.

Therefore, the correct choice is:

$x = \sqrt{3} - 2, -\sqrt{3} - 2$