Factor. \[ 5 z^{2}-22 z+8 \] \( \square \)

To factor the quadratic expression \( 5z^2 - 22z + 8 \), we need to look for two numbers that multiply to \( 5 \times 8 = 40 \) and add up to \( -22 \). The numbers \( -20 \) and \( -2 \) do the trick! So, we can rewrite the middle term: \[ 5z^2 - 20z - 2z + 8 \] Now, we group the terms: \[ (5z^2 - 20z) + (-2z + 8) \] Factoring each group gives us: \[ 5z(z - 4) - 2(z - 4) \] Now, we can factor out the common factor \( (z - 4) \): \[ (z - 4)(5z - 2) \] And there you have it! The factored form is \( (z - 4)(5z - 2) \). As a fun fact, factoring quadratics is like solving a puzzle—finding the right numbers to fit the equation! With practice, it becomes second nature. In the grand scheme of math, quadratics often pop up in real-world scenarios, like calculating the trajectory of a thrown ball or optimizing areas in agriculture. Knowing how to factor them can provide insights into these practical situations!