Factor completely. 2w^2+16w+14 square

To factor the quadratic expression \(2w^2 + 16w + 14\) completely, we can follow these steps:<br /><br />1. **Factor out the greatest common factor (GCF):**<br /> The GCF of the coefficients \(2, 16,\) and \(14\) is \(2\). So, we factor out \(2\):<br /> \[<br /> 2(w^2 + 8w + 7)<br /> \]<br /><br />2. **Factor the quadratic expression inside the parentheses:**<br /> We need to factor \(w^2 + 8w + 7\). We look for two numbers that multiply to \(7\) (the constant term) and add up to \(8\) (the coefficient of the linear term).<br /><br /> The numbers \(1\) and \(7\) satisfy these conditions because:<br /> \[<br /> 1 \cdot 7 = 7 \quad \text{and} \quad 1 + 7 = 8<br /> \]<br /><br />3. **Write the factored form:**<br /> Using the numbers \(1\) and \(7\), we can write the quadratic as:<br /> \[<br /> w^2 + 8w + 7 = (w + 1)(w + 7)<br /> \]<br /><br />4. **Include the GCF:**<br /> Finally, we include the GCF that we factored out initially:<br /> \[<br /> 2(w + 1)(w + 7)<br /> \]<br /><br />So, the completely factored form of \(2w^2 + 16w + 14\) is:<br />\[<br />\boxed{2(w + 1)(w + 7)}<br />\]