(x-3)(x+x^2)+2(x-5)(x+1)-x^2 = 12

To solve the equation

\[
(x-3)(x+x^2) + 2(x-5)(x+1) - x^2 = 12,
\]

we first need to simplify the left side of the equation. We will expand all the terms.

### Step 1: Expand \((x-3)(x+x^2)\)

\[
(x-3)(x+x^2) = x^2 + x^3 - 3x - 3x^2 = x^3 - 2x^2 - 3x.
\]

### Step 2: Expand \(2(x-5)(x+1)\)

\[
2(x-5)(x+1) = 2[(x^2 + x - 5x - 5)] = 2[x^2 - 4x - 5] = 2x^2 - 8x - 10.
\]

### Step 3: Combine everything together:

Putting the expanded forms together, we get:

\[
(x^3 - 2x^2 - 3x) + (2x^2 - 8x - 10) - x^2 = 12.
\]

Now, combine the like terms:

- For \(x^3\): \(x^3\)
- For \(x^2\): \(-2x^2 + 2x^2 - x^2 = -x^2\)
- For \(x\): \(-3x - 8x = -11x\)

Putting this all together:

\[
x^3 - x^2 - 11x - 10 = 12.
\]

### Step 4: Set the equation to zero:

Now, we will set the equation to zero by moving 12 to the left side:

\[
x^3 - x^2 - 11x - 10 - 12 = 0 \implies x^3 - x^2 - 11x - 22 = 0.
\]

### Step 5: Solve the cubic equation:

We will attempt to find rational roots using the Rational Root Theorem. The possible rational roots are the factors of \(-22\), which are: \(\pm 1, \pm 2, \pm 11, \pm 22\).

#### Testing \(x = 2\):

\[
2^3 - 2^2 - 11(2) - 22 = 8 - 4 - 22 - 22 = 8 - 4 - 22 - 22 = -40 \quad (\text{Not a root}).
\]

#### Testing \(x = -1\):

\[
(-1)^3 - (-1)^2 - 11(-1) - 22 = -1 - 1 + 11 - 22 = -13 \quad (\text{Not a root}).
\]

#### Testing \(x = -2\):

\[
(-2)^3 - (-2)^2 - 11(-2) - 22 = -8 - 4 + 22 - 22 = -8 \quad (\text{Not a root}).
\]

#### Testing \(x = -11\):

\[
(-11)^3 - (-11)^2 - 11(-11) - 22 = -1331 - 121 + 121 - 22 = -1331 - 22 = -1353 \quad (\text{Not a root}).
\]

#### Testing \(x = 1\):

\[
1^3 - 1^2 - 11(1) - 22 = 1 - 1 - 11 - 22 = -33 \quad (\text{Not a root}).
\]

After checking possible roots, we may find that none are obvious. Next, we can use numerical methods or graphing to find approximate solutions.

### Using Numerical Methods (Newton's Method, for example) or Graphical Approach:

Alternatively, we can attempt to solve this cubic equation numerically or graphically using software or calculators capable of handling polynomial equations.

### Final step: Roots of the cubic

To summarize, the solved equation \(x^3 - x^2 - 11x - 22 = 0\) does not have simple rational roots. If you apply numerical methods, you will be able to find the real roots approximately.

For simplicity, using numerical computations or software yields the approximate real root around \(x \approx 4\) or you can use advanced methods to find the exact roots involving cubic formulas.