Factor the following expression completely: 2y^2+9y-35=square

To factor the quadratic expression \(2y^2 + 9y - 35\) completely, we need to find two binomials whose product gives us the original quadratic expression.<br /><br />First, we identify the coefficients:<br />- \(a = 2\)<br />- \(b = 9\)<br />- \(c = -35\)<br /><br />We need to find two numbers that multiply to \(a \cdot c = 2 \cdot (-35) = -70\) and add up to \(b = 9\).<br /><br />Let's denote these two numbers as \(m\) and \(n\). So, we have:<br />\[ m \cdot n = -70 \]<br />\[ m + n = 9 \]<br /><br />By examining the pairs of factors of \(-70\), we find that \(m = 14\) and \(n = -5\) satisfy these conditions because:<br />\[ 14 \cdot (-5) = -70 \]<br />\[ 14 + (-5) = 9 \]<br /><br />Next, we rewrite the middle term \(9y\) using \(m\) and \(n\):<br />\[ 2y^2 + 9y - 35 = 2y^2 + 14y - 5y - 35 \]<br /><br />Now, we group the terms in pairs:<br />\[ 2y^2 + 14y - 5y - 35 = (2y^2 + 14y) + (-5y - 35) \]<br /><br />We factor out the greatest common factor (GCF) from each pair:<br />\[ = 2y(y + 7) - 5(y + 7) \]<br /><br />We notice that \((y + 7)\) is a common factor:<br />\[ = (2y - 5)(y + 7) \]<br /><br />Thus, the completely factored form of the expression \(2y^2 + 9y - 35\) is:<br />\[ \boxed{(2y - 5)(y + 7)} \]