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When we start factoring any expression, it's essential to find the **Greatest Common Factor (GCF)**. The GCF is the largest factor that divides all terms of the expression evenly. In the trinomial \(10b^6 - 19b^4 + 6b^2\), we examine each term to identify any common factors.
Every term in the given trinomial shares a common factor of \(b^2\), so \(b^2\) is the GCF. Factoring out \(b^2\) simplifies the trinomial by reducing each exponent by 2. We get:

• Original Trinomial: \(10b^6\) - \(19b^4\) + \(6b^2\)
• Factored by GCF: \(b^2(10b^4 - 19b^2 + 6)\)

Identifying and factoring out the GCF is crucial as it simplifies the expression, making further factoring possible.

Once we factor out the GCF, we often need further techniques to simplify the remaining expression. Factoring by grouping is a strategic approach, especially useful for quadratic expressions after dealing with the GCF.
After replacing the variables in terms that are easier to handle (like letting \(x = b^2\)), we convert our expression to \(10x^2 - 19x + 6\). Now, we can use factoring by grouping:

• Split the middle term using numbers that multiply to a(x) \(\cdot\) c: Here, \(10 \cdot 6 = 60\). Two numbers that work are \(-15\) and \(-4\) (adding to \(-19\))
• Rewrite the expression: \(10x^2 - 15x - 4x + 6\)
• Group and factor: \((10x^2 - 15x) + (-4x + 6)\)
• Factor out the GCF from each group: \(5x(2x - 3) - 2(2x - 3)\)
• Notice the common factor: \((2x - 3)\), leading to \((5x - 2)(2x - 3)\)

Factoring by grouping is a clever way to split a quadratic so that we can spot and extract common factors, leading us to the final factorization step.

Quadratic expressions are typically in the form \(ax^2 + bx + c\). These expressions are foundational in algebra and can often be factored down into binomials. In our case, after substituting \(x = b^2\), the expression changes to a standard quadratic form \(10x^2 - 19x + 6\).
To factor a quadratic expression, we identify two numbers that solve:

• Multiply to \(a \cdot c\)
• Add to \(b\)

Here, we need numbers that multiply to \(60\) (from \(10 \times 6\)) and add to \(-19\), which are \(-15\) and \(-4\). Rewriting and grouping these terms helps break the quadratic into factors.
After successful grouping and factoring, substituting back the original variables (in terms of \(b\)) gives us the complete factorization:

• \((5b^2 - 2)(2b^2 - 3)\)

Quadratics involve both skilful insight to recognize patterns and precise steps to make successful simplification, making them a central concept in algebra.