Solve for a, b, c, and d: Balancing Quadratic and Radical Expressions

To solve this problem, we'll proceed with the following steps:

• Step 1: Expand all squared binomials using the formula $(A + B)^2 = A^2 + 2AB + B^2$.
• Step 2: Balance the coefficients from both sides of the equation.
• Step 3: Formulate equations by comparing coefficients and solve for $a$, $b$, $c$, and $d$.

Let's go through these steps:

Step 1:

Expanding the left side:

Thus, the left side becomes:

Expanding the right side:

The right side simplifies to:

Step 2:

Equate coefficients of like powers of $x$:

Equated constant terms give:

Step 3:

Solving the obtained equations yields:

Therefore, the solution to this problem is proven correct and matches choice 3: $a=-(64+3\sqrt{6}), b=24+\sqrt{6}, c=1, d=\sqrt{6}$.