Factor the following expression. \[ \begin{array}{l}8 x^{2}+22 x+5 \\ (4 x+[?])(\square x+\square)\end{array} \]

To factor the expression \(8 x^{2} + 22 x + 5\), we look for two binomials in the form \((4 x + a)(2 x + b)\). 1. The first term in both binomials naturally leads us to choose \(4x\) from the first and \(2x\) from the second, since \(4x \cdot 2x = 8x^2\). 2. Next, we need to find \(a\) and \(b\) such that: - \(4b + 2a = 22\) (the coefficient of \(x\)) - \(ab = 5\) (the constant term) We know, from potential integer pairs whose product is 5, that \(a\) and \(b\) could be \((1,5)\) or \((5,1)\). Let's test \(a = 1\) and \(b = 5\): - \(4(5) + 2(1) = 20 + 2 = 22\), which satisfies the x-term condition. Thus, we have found suitable values: The factored form is: \[(4x + 1)(2x + 5).\]