Question

1. 81x² + 4

Explanation:

Step1: Recall the formula for sum of squares

The expression is \(81x^2 + 4\). We can rewrite it as \((9x)^2+2^2\). But the sum of squares \(a^2 + b^2\) can be factored in the complex number system using the formula \(a^2 + b^2=(a + bi)(a - bi)\), where \(i\) is the imaginary unit with \(i^2=- 1\).

Step2: Apply the formula

Here, \(a = 9x\) and \(b = 2\). So, \(81x^2+4=(9x)^2 + 2^2=(9x + 2i)(9x-2i)\)

Answer:

\((9x + 2i)(9x - 2i)\) (in complex numbers) or it is irreducible over the real numbers. If we consider real - number factoring, it cannot be factored further using real coefficients. If complex - number factoring is allowed, the factored form is \((9x + 2i)(9x - 2i)\)