Solve the Equation: 7 = 5x² + 8x + (x+4)² to Find X

To solve this quadratic equation, follow the steps below:

• Step 1: Begin by expanding $(x+4)^2$.
• Step 2: Expand to get $(x+4)^2 = x^2 + 8x + 16$.
• Step 3: Substitute the expanded form into the original equation: $7 = 5x^2 + 8x + x^2 + 8x + 16$.\
• Step 4: Combine like terms: $7 = 6x^2 + 16x + 16$.
• Step 5: Rearrange into standard quadratic form: $6x^2 + 16x + 9 = 0$.
• Step 6: Use the quadratic formula $x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}$, where $a = 6$, $b = 16$, and $c = 9$.
• Step 7: Compute the discriminant: $b^2 - 4ac = 16^2 - 4(6)(9) = 256 - 216 = 40$.
• Step 8: Substitute into the quadratic formula: $x = \frac{{-16 \pm \sqrt{40}}}{12} = \frac{{-16 \pm 2\sqrt{10}}}{12} = -\frac{4}{3} \pm \frac{\sqrt{10}}{6}$.

Thus, the solutions are $x = -\frac{4}{3} + \frac{\sqrt{10}}{6}$ and $x = -\frac{4}{3} - \frac{\sqrt{10}}{6}$.

Therefore, the correct solution, corresponding to the provided choices, is $-\frac{4}{3} \pm \frac{\sqrt{10}}{6}$.