The given expression is:

$\left( \sqrt{3} + 4 \right)^4 + \left( \sqrt{3} - 1 \right)^4 = 56$

To solve this using the binomial theorem, we need to expand both terms separately. The binomial expansion for $(a + b)^n$ is given by:

$(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$

Step 1: Expand $(\sqrt{3} + 4)^4$

For the term $(\sqrt{3} + 4)^4$, we apply the binomial expansion with $a = \sqrt{3}$, $b = 4$, and $n = 4$:

$(\sqrt{3} + 4)^4 = \binom{4}{0} (\sqrt{3})^4 (4)^0 + \binom{4}{1} (\sqrt{3})^3 (4)^1 + \binom{4}{2} (\sqrt{3})^2 (4)^2 + \binom{4}{3} (\sqrt{3})^1 (4)^3 + \binom{4}{4} (\sqrt{3})^0 (4)^4$

Let's simplify this:

• $\binom{4}{0} = 1$, $(\sqrt{3})^4 = 9$, $(4)^0 = 1$
• $\binom{4}{1} = 4$, $(\sqrt{3})^3 = 3\sqrt{3}$, $(4)^1 = 4$
• $\binom{4}{2} = 6$, $(\sqrt{3})^2 = 3$, $(4)^2 = 16$
• $\binom{4}{3} = 4$, $(\sqrt{3})^1 = \sqrt{3}$, $(4)^3 = 64$
• $\binom{4}{4} = 1$, $(\sqrt{3})^0 = 1$, $(4)^4 = 256$

Substituting these values:

$(\sqrt{3} + 4)^4 = 1 \cdot 9 + 4 \cdot 3\sqrt{3} \cdot 4 + 6 \cdot 3 \cdot 16 + 4 \cdot \sqrt{3} \cdot 64 + 1 \cdot 256$

Simplifying the terms:

$(\sqrt{3} + 4)^4 = 9 + 48\sqrt{3} + 288 + 256\sqrt{3} + 256$

$(\sqrt{3} + 4)^4 = 553 + 304\sqrt{3}$

Step 2: Expand $(\sqrt{3} - 1)^4$

Similarly, for the second term $(\sqrt{3} - 1)^4$, we use $a = \sqrt{3}$, $b = -1$, and $n = 4$. The expansion is:

$(\sqrt{3} - 1)^4 = \binom{4}{0} (\sqrt{3})^4 (-1)^0 + \binom{4}{1} (\sqrt{3})^3 (-1)^1 + \binom{4}{2} (\sqrt{3})^2 (-1)^2 + \binom{4}{3} (\sqrt{3})^1 (-1)^3 + \binom{4}{4} (\sqrt{3})^0 (-1)^4$

Substitute the same binomial coefficients:

• $\binom{4}{0} = 1$, $(\sqrt{3})^4 = 9$, $(-1)^0 = 1$
• $\binom{4}{1} = 4$, $(\sqrt{3})^3 = 3\sqrt{3}$, $(-1)^1 = -1$
• $\binom{4}{2} = 6$, $(\sqrt{3})^2 = 3$, $(-1)^2 = 1$
• $\binom{4}{3} = 4$, $(\sqrt{3})^1 = \sqrt{3}$, $(-1)^3 = -1$
• $\binom{4}{4} = 1$, $(\sqrt{3})^0 = 1$, $(-1)^4 = 1$

Substituting:

$(\sqrt{3} - 1)^4 = 1 \cdot 9 - 4 \cdot 3\sqrt{3} \cdot 1 + 6 \cdot 3 \cdot 1 - 4 \cdot \sqrt{3} \cdot 1 + 1 \cdot 1$

Simplifying the terms:

$(\sqrt{3} - 1)^4 = 9 - 12\sqrt{3} + 18 - 4\sqrt{3} + 1$

$(\sqrt{3} - 1)^4 = 28 - 16\sqrt{3}$

Step 3: Add the Two Expansions

Now, we add the two expanded terms:

$(\sqrt{3} + 4)^4 + (\sqrt{3} - 1)^4 = (553 + 304\sqrt{3}) + (28 - 16\sqrt{3})$

Simplifying:

$553 + 28 + (304\sqrt{3} - 16\sqrt{3}) = 581 + 288\sqrt{3}$

Step 4: Solve for the Equation

The given equation is:

$(\sqrt{3} + 4)^4 + (\sqrt{3} - 1)^4 = 56$

However, we have:

$581 + 288\sqrt{3} = 56$

This does not match the equation, indicating that something went wrong or additional context is needed. Would you like me to clarify or check for other details?