Question 8 Factor completely: \( 3 x^{2}+7 x+4 \) Factored Expression \( =\underline{\text { Blank } 1} \)

To factor the quadratic expression \( 3x^2 + 7x + 4 \) completely, we look for two numbers that multiply to \( 3 \times 4 = 12 \) and add to \( 7 \). Those numbers are \( 3 \) and \( 4 \). We can then rewrite the equation as: \[ 3x^2 + 3x + 4x + 4 \] Now, grouping the terms gives us: \[ (3x^2 + 3x) + (4x + 4) \] Factoring out the common factors from each group results in: \[ 3x(x + 1) + 4(x + 1) \] Since \( (x + 1) \) is a common factor, we factor it out: \[ (3x + 4)(x + 1) \] So, the completely factored expression is: \[ \underline{(3x + 4)(x + 1)} \] --- Did you know that factoring quadratics like this goes back to ancient civilizations? The Babylonians were among the first to solve quadratic equations, using geometric methods long before algebra was formalized. They had some serious math skills for their time! For real-world application, factoring quadratics can help in many practical scenarios, like determining the projectile motion of objects. If you are designing a rollercoaster, for instance, understanding the factors can help optimize the ride's height and speed, ensuring safety while maximizing thrill!