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When you come across two binomials, like

• \((a - b)\)
• \((a + b)\)

they form what is known as a difference of squares.
This occurs because their multiplication simplifies into a particular pattern: \((a - b)(a + b) = a^2 - b^2\). This is called the difference of squares because you're literally taking the square of the first term, and subtracting the square of the second term.

In this example,

• \( a = 3\sqrt{x} \) and
• \( b = 4 \).

So, when you plug these into the formula \( (a - b)(a + b) = a^2 - b^2 \), you get

• \( (3\sqrt{x})^2 - (4)^2 = 9x - 16 \).

The difference of squares is a crucial concept for simplifying expressions like this. It helps quickly transform complex products into simpler expressions.

Understanding square roots is fundamental in algebra.
The square root of a number is a value that, when multiplied by itself, gives the original number.
In mathematical notation, the square root symbol is \( \sqrt{} \). For instance, \( \sqrt{x} \) produces a number which, when squared, results in \( x \).

When dealing with algebraic terms, such as \( 3\sqrt{x} \), the number \( 3 \) is a coefficient that can scale the value of the square root.

• Here, \( 3\sqrt{x} \) means multiplying the square root of \( x \) by 3.

This ability to manipulate square roots forms a base for more complex algebraic operations.

Square roots add an additional layer when working with algebra since they represent values that are not straightforward to visualize. In our exercise, recognizing \( 3\sqrt{x} \) and correctly squaring it helps in effective simplification.

Multiplying binomials is a common task in algebra, and it allows you to expand expressions. A binomial is simply a polynomial with two terms.
When you multiply two binomials, like

• \((a - b)\) and
• \((a + b)\),

you follow a systematic approach which often helps visualize the resulting expression.

The product of these specific binomials results in the difference of squares, discussed earlier. It simplifies evaluation and reduces error risk by sticking to a formulaic approach instead of independently multiplying each component.

• For example, \((3\sqrt{x} - 4)(3\sqrt{x} + 4) \) automatically simplifies to \((3\sqrt{x})^2 - 4^2 \).

Thus, binomial multiplication doesn't just assist in calculation; it acts as a foundation for understanding more significant polynomial operations. Recognizing these structures and applying formulaic shortcuts leads to efficient solutions and deeper algebraic insight.