4. Factored completely, m^5+m^3-6m is equivalent to

To factor the expression \( m^5 + m^3 - 6m \) completely, we can follow these steps:<br /><br />1. **Factor out the greatest common factor (GCF):**<br /> The GCF of the terms \( m^5 \), \( m^3 \), and \( -6m \) is \( m \).<br /><br /> \[<br /> m^5 + m^3 - 6m = m(m^4 + m^2 - 6)<br /> \]<br /><br />2. **Factor the quadratic expression inside the parentheses:**<br /> We need to factor \( m^4 + m^2 - 6 \). Let's set \( x = m^2 \). Then the expression becomes a quadratic in terms of \( x \):<br /><br /> \[<br /> x^2 + x - 6<br /> \]<br /><br />3. **Factor the quadratic expression:**<br /> We look for two numbers that multiply to \(-6\) and add up to \(1\). These numbers are \(3\) and \(-2\).<br /><br /> \[<br /> x^2 + x - 6 = (x + 3)(x - 2)<br /> \]<br /><br />4. **Substitute back \( x = m^2 \):**<br /> Replace \( x \) with \( m^2 \) in the factored form:<br /><br /> \[<br /> (m^2 + 3)(m^2 - 2)<br /> \]<br /><br />5. **Combine with the GCF factored out initially:**<br /> Finally, we include the \( m \) that we factored out initially:<br /><br /> \[<br /> m(m^2 + 3)(m^2 - 2)<br /> \]<br /><br />So, the completely factored form of \( m^5 + m^3 - 6m \) is:<br /><br />\[<br />m(m^2 + 3)(m^2 - 2)<br />\]