1. \( 3 x^{2}+11 x+10=(3 x+5)(x-2) \)

Let's first expand the right side of the equation: \[ (3x + 5)(x - 2) = 3x^2 - 6x + 5x - 10 = 3x^2 - x - 10. \] Now, we have \( 3x^2 + 11x + 10 = 3x^2 - x - 10 \). Subtract \( 3x^2 \) from both sides: \[ 11x + 10 = -x - 10. \] Next, we can add \( x \) to both sides: \[ 12x + 10 = -10. \] Then, subtract 10 from both sides: \[ 12x = -20. \] Finally, divide both sides by 12: \[ x = -\frac{20}{12} = -\frac{5}{3}. \] So, the solution to the equation is \( x = -\frac{5}{3} \). Now, let’s work out another fun mathematical exploration with this problem! This kind of equation demonstrates the magic of factoring and solving quadratics. By rewriting a polynomial into factored form, we can easily find the roots, which is immensely helpful in fields like physics, engineering, and even economics when we analyze models or curves! Equally fascinating is that polynomial equations can have at most as many roots as their degree. In this case, since it's a quadratic (degree 2), we can expect two solutions, but we found just one real root due to its factorization. This hints at how deeper analyses can reveal the nature of roots—whether they are real, complex, repeated, or distinct!