Factor: \( 3 x^{2}+17 x+10 \)

To factor the quadratic \( 3x^{2} + 17x + 10 \), we start by looking for two numbers that multiply to \( 3 \times 10 = 30 \) and add up to \( 17 \). Those two numbers are \( 15 \) and \( 2 \). Rewrite the middle term: \( 3x^{2} + 15x + 2x + 10 \). Next, group the terms: \( (3x^{2} + 15x) + (2x + 10) \). Factor each group: \( 3x(x + 5) + 2(x + 5) \). Now, factor out the common factor \( (x + 5) \): \( (3x + 2)(x + 5) \). So, the factored form is \( (3x + 2)(x + 5) \). To check our work, we can quickly multiply the factors back together to see if we return to the original expression. And voilà! There's the fun of factorization wrapped up in this tidy product!