To solve the equation \(9x^2 + 14x + 4 = \sqrt{x + 2}\), we need to eliminate the square root by squaring both sides of the equation. However, we must be careful to check for extraneous solutions that can arise from this process.

1. **Square both sides:**

\[
(9x^2 + 14x + 4)^2 = (\sqrt{x + 2})^2
\]

Simplifying the right-hand side:

\[
(9x^2 + 14x + 4)^2 = x + 2
\]

2. **Expand the left-hand side:**

\[
(9x^2 + 14x + 4)^2 = (9x^2 + 14x + 4)(9x^2 + 14x + 4)
\]

Use the distributive property (FOIL method):

\[
(9x^2 + 14x + 4)(9x^2 + 14x + 4) = 81x^4 + 126x^3 + 36x^2 + 126x^3 + 196x^2 + 56x + 36x^2 + 56x + 16
\]

Combine like terms:

\[
81x^4 + 252x^3 + 268x^2 + 112x + 16
\]

3. **Set the equation equal to \(x + 2\):**

\[
81x^4 + 252x^3 + 268x^2 + 112x + 16 = x + 2
\]

4. **Move all terms to one side to set the equation to zero:**

\[
81x^4 + 252x^3 + 268x^2 + 112x + 16 - x - 2 = 0
\]

Simplify:

\[
81x^4 + 252x^3 + 268x^2 + 111x + 14 = 0
\]

5. **Solve the polynomial equation:**

Solving a quartic equation analytically can be very complex, and typically numerical methods or computational tools are used. However, we can check for possible rational roots using the Rational Root Theorem, which states that any rational solution, \(p/q\), is a factor of the constant term (14) divided by a factor of the leading coefficient (81).

Possible rational roots are: \(\pm 1, \pm 2, \pm 7, \pm 14, \pm 1/3, \pm 2/3, \pm 7/3, \pm 14/3, \pm 1/9, \pm 2/9, \pm 7/9, \pm 14/9\).

Testing these potential roots by substitution or using synthetic division can help identify any rational solutions.

6. **Verify potential solutions:**

After finding potential solutions, substitute them back into the original equation \(9x^2 + 14x + 4 = \sqrt{x + 2}\) to verify if they are valid and not extraneous.

Given the complexity of solving quartic equations by hand, it is often more practical to use numerical methods or a computational tool to find approximate solutions.