To solve the equation \(x^2 + 3x + 8 = 2(x + 1)\sqrt{x + 7}\), we can start by isolating the square root term. Here are the steps:

1. **Isolate the square root term:**
\[
x^2 + 3x + 8 = 2(x + 1)\sqrt{x + 7}
\]

2. **Square both sides to eliminate the square root:**
\[
\left(x^2 + 3x + 8\right)^2 = \left[2(x + 1)\sqrt{x + 7}\right]^2
\]
\[
\left(x^2 + 3x + 8\right)^2 = 4(x + 1)^2(x + 7)
\]

3. **Expand both sides:**

Left side:
\[
(x^2 + 3x + 8)^2 = x^4 + 6x^3 + 25x^2 + 48x + 64
\]

Right side:
\[
4(x + 1)^2(x + 7) = 4(x^2 + 2x + 1)(x + 7)
\]
\[
= 4(x^3 + 9x^2 + 21x + 7)
\]
\[
= 4x^3 + 36x^2 + 84x + 28
\]

4. **Set the expanded forms equal to each other:**
\[
x^4 + 6x^3 + 25x^2 + 48x + 64 = 4x^3 + 36x^2 + 84x + 28
\]

5. **Move all terms to one side to set the equation to zero:**
\[
x^4 + 6x^3 + 25x^2 + 48x + 64 - 4x^3 - 36x^2 - 84x - 28 = 0
\]
\[
x^4 + 2x^3 - 11x^2 - 36x + 36 = 0
\]

6. **Solve the polynomial equation:**

This is a quartic equation, and solving it analytically can be complex. We can try to find rational roots using the Rational Root Theorem, which states that any rational solution, \(p/q\), is a factor of the constant term (36) divided by a factor of the leading coefficient (1).

Possible rational roots are \( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 9, \pm 12, \pm 18, \pm 36 \).

Testing these values:

- \(x = 1\):
\[
1^4 + 2(1)^3 - 11(1)^2 - 36(1) + 36 = 1 + 2 - 11 - 36 + 36 = -8 \quad (\text{not a root})
\]

- \(x = -1\):
\[
(-1)^4 + 2(-1)^3 - 11(-1)^2 - 36(-1) + 36 = 1 - 2 - 11 + 36 + 36 = 60 \quad (\text{not a root})
\]

- \(x = 2\):
\[
2^4 + 2(2)^3 - 11(2)^2 - 36(2) + 36 = 16 + 16 - 44 - 72 + 36 = -48 \quad (\text{not a root})
\]

- \(x = -2\):
\[
(-2)^4 + 2(-2)^3 - 11(-2)^2 - 36(-2) + 36 = 16 - 16 - 44 + 72 + 36 = 64 \quad (\text{not a root})
\]

- \(x = 3\):
\[
3^4 + 2(3)^3 - 11(3)^2 - 36(3) + 36 = 81 + 54 - 99 - 108 + 36 = -36 \quad (\text{not a root})
\]

- \(x = -3\):
\[
(-3)^4 + 2(-3)^3 - 11(-3)^2 - 36(-3) + 36 = 81 - 54 - 99 + 108 + 36 = 72 \quad (\text{not a root})
\]

- \(x = 4\):
\[
4^4 + 2(4)^3 - 11(4)^2 - 36(4) + 36 = 256 + 128 - 176 - 144 + 36 = 100 \quad (\text{not a root})
\]

- \(x = -4\):
\[
(-4)^4 + 2(-4)^3 - 11(-4)^2 - 36(-4) + 36 = 256 - 128 - 176 + 144 + 36 = 132 \quad (\text{not a root})
\]

- \(x = 6\):
\[
6^4 + 2(6)^3 - 11(6)^2 - 36(6) + 36 = 1296 + 432 - 396 - 216 + 36 = 1152 \quad (\text{not a root})
\]

- \(x = -6\):
\[
(-6)^4 + 2(-6)^3 - 11(-6)^2 - 36(-6) + 36 = 1296 - 432 - 396 + 216 + 36 = 720 \quad (\text{not a root})
\]

- \(x = 9\):
\[
9^4 + 2(9)^3 - 11(9)^2 - 36(9) + 36 = 6561 + 1458 - 891 - 324 + 36 = 6840 \quad (\text{not a root})
\]

- \(x = -9\):
\[
(-9)^4 + 2(-9)^3 - 11(-9)^2 - 36(-9) + 36 = 6561 - 1458 - 891 + 324 + 36 = 4572 \quad (\text{not a root})
\]

- \(x = 12\):
\[
12^4 + 2(12)^3 - 11(12)^2 - 36(12) + 36 = 20736 + 3456 - 1584 - 432 + 36 = 22008 \quad (\text{not a root})
\]

- \(x = -12\):
\[
(-12)^4 + 2(-12)^3 - 11(-12)^2 - 36(-12) + 36 = 20736 - 3456 - 1584 + 432 + 36 = 16164 \quad (\text{not a root})
\]

- \(x = 18\):
\[
18^4 + 2(18)^3 - 11(18)^2 - 36(18) + 36 = 104976 + 11664 - 3564 - 648 + 36 = 112464 \quad (\text{not a root})
\]

- \(x = -18\):
\[
(-18)^4 + 2(-18)^3 - 11(-18)^2 - 36(-18) + 36 = 104976 - 11664 - 3564 + 648 + 36 = 90432 \quad (\text{not a root})
\]

- \(x = 36\):
\[
36^4 + 2(36)^3 - 11(36)^2 - 36(36) + 36 = 1679616 + 93312 - 14256 - 1296 + 36 = 1763412 \quad (\text{not a root})
\]

- \(x = -36\):
\[
(-36)^4 + 2(-36)^3 - 11(-36)^2 - 36(-36) + 36 = 1679616 - 93312 - 14256 + 1296 + 36 = 1579380 \quad (\text{not a root})
\]

Since none of the rational roots work, the polynomial might have irrational or complex roots. Solving this quartic equation analytically is quite complex, so it might be more practical to use numerical methods or a computer algebra system to find the roots.

Alternatively, we can check if there are any simpler methods or substitutions that might simplify the equation, but given the complexity, numerical methods are often the most straightforward approach.