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When multiplying a binomial, it's vital to remember the structure of the expression. A binomial contains two terms, like \(3 + \sqrt{7}\). Multiplying binomials involves using the distributive property to ensure every term is accounted for.

For a more general form, consider \(a + b\), where \(a\) and \(b\) represent any numbers or variables. To multiply \((a + b)^{2}\), you need to apply the distributive property to write it as:

\( (a + b)(a + b) = a(a + b) + b(a + b)\).

Distributing, we get: \(a^{2} + ab + ba + b^{2}\ = a^{2} + 2ab + b^{2}\). This follows a general pattern known as the binomial theorem. Breaking it down step-by-step ensures that all components are multiplied correctly.

Squaring a binomial is a common task in algebra. Squared expressions follow the identity: \((a + b)^{2} = a^{2} + 2ab + b^{2}\).

This means you multiply the binomial by itself, leading to three different terms: the square of each component and twice the product of both components.

Let's use our original example \(3 + \sqrt{7}\). We know: \(a = 3\) and \(b = \sqrt{7}\). The squared formula becomes: \((3 + \sqrt{7})^{2} = 3^{2} + 2(3)(\sqrt{7}) + (\sqrt{7})^{2}\).

Calculating these:
\(3^{2} = 9\),
\[2(3)(\sqrt{7}) = 6 \sqrt{7}\],
and \(\sqrt{7}^{2} = 7\).

Adding everything together gives us: \9 + 6 \sqrt{7} + 7\.

Algebraic expansion uses rules to systematically break down and simplify expressions. When dealing with binomials, using the binomial theorem formula simplifies the process.

For our specific problem, \((3+\sqrt{7})^{2} \), applying the binomial theorem simplifies the expansion: \((a + b)^{2} = a^{2} + 2ab + b^{2}\).

This ensures that each term is accounted for and accurately calculated.

First, square the individual components: 3 and \sqrt{7}\.
Next, multiply both terms together and then by two: \(3^{2} = 9\), \(2 \cdot 3 \cdot \sqrt{7} = 6 \sqrt{7}\), and \((\sqrt{7})^{2} = 7\).

Finally, add the results for a final expression of: \9 + 6 \sqrt{7} + 7\. Simplify, combining like terms to get: \16 + 6 \sqrt{7}\.