If `|[2x, x+3], [2(x+1), x+1]| = |[1, 5], [3, 3]|`, find the value of x.

Explanation

To solve the equation given by the determinants, we start with the expression:

\[ \begin{vmatrix} 2x & x + 3 \\ 2(x + 1) & x + 1 \end{vmatrix} = \begin{vmatrix} 1 & 5 \\ 3 & 3 \end{vmatrix} \]

### Step 1: Calculate the determinant on the left side

The determinant of a 2x2 matrix \(\begin{vmatrix} a & b \\ c & d \end{vmatrix}\) is calculated as \(ad - bc\).

For the left side:

\[ \text{Determinant} = (2x)(x + 1) - (x + 3)(2(x + 1)) \]

### Step 2: Expand the terms

Now we will expand both terms:

1. \( (2x)(x + 1) = 2x^2 + 2x \)
2. \( (x + 3)(2(x + 1)) = (x + 3)(2x + 2) = 2x^2 + 2x + 6x + 6 = 2x^2 + 8x + 6 \)

### Step 3: Substitute back into the determinant equation

Now substituting back into the determinant equation:

\[ 2x^2 + 2x - (2x^2 + 8x + 6) = 0 \]

### Step 4: Simplify the equation

Distributing the negative sign:

\[ 2x^2 + 2x - 2x^2 - 8x - 6 = 0 \]

This simplifies to:

\[ -6x - 6 = 0 \]

### Step 5: Solve for x

Now, we can solve for \(x\):

\[ -6x = 6 \implies x = -1 \]

### Step 6: Verify with the right side determinant

Now, we calculate the right side determinant:

\[ \begin{vmatrix} 1 & 5 \\ 3 & 3 \end{vmatrix} = (1)(3) - (5)(3) = 3 - 15 = -12 \]

### Conclusion

Thus, we have:

\[ 2(x + 1) = -12 \implies x + 1 = -6 \implies x = -7 \]

So the value of \(x\) is:

\[ \boxed{-7} \]