Question

factor the polynomial completely. $2x^{3}+12x^{2}+14x=\square$

Explanation:

Step1: Find the GCF

The greatest common factor (GCF) of \(2x^3\), \(12x^2\), and \(14x\) is \(2x\). Factor out \(2x\) from each term.
\(2x^3 + 12x^2 + 14x = 2x(x^2 + 6x + 7)\)

Step2: Check if quadratic factors

The quadratic \(x^2 + 6x + 7\) has discriminant \(b^2 - 4ac = 6^2 - 4(1)(7) = 36 - 28 = 8\), which is not a perfect square, so it doesn't factor over the integers. So the complete factorization is \(2x(x^2 + 6x + 7)\).

Answer:

\(2x(x^2 + 6x + 7)\)