7. \( 6 x^{2}+26 x+8 \)

To factor the quadratic expression \( 6 x^{2}+26 x+8 \), start by identifying the coefficients: \( a = 6 \), \( b = 26 \), and \( c = 8 \). Use the factoring by grouping method or the quadratic formula to find the roots. If you simplify by dividing all terms by 2 first, you get \( 3x^2 + 13x + 4 \). This can then be factored as \( (3x + 1)(x + 4) \). Don't forget to include the factor of 2 when you rewrite the original expression! In real-world terms, quadratic expressions like this one might represent various phenomena, such as projectile motion in physics or even profit maximization in business. For example, if you track the revenue generated by selling \( x \) units of a product, this expression could model how profit changes based on production levels, making it crucial for making data-driven decisions and optimizing returns!