Use factorisation to solve the equation. \[ 3 p^{2}+p-10=0 \]

To solve the equation \( 3p^{2}+p-10=0 \) using factorization, we start by looking for two numbers that multiply to \( 3 \times -10 = -30 \) and add up to \( 1 \) (the coefficient of \( p \)). The numbers \( 6 \) and \( -5 \) work since \( 6 \times (-5) = -30 \) and \( 6 + (-5) = 1 \). We can rewrite the middle term and factor the quadratic: \[ 3p^{2} + 6p - 5p - 10 = 0 \] Now, group the terms: \[ (3p^{2} + 6p) + (-5p - 10) = 0 \] Factor out the common terms: \[ 3p(p + 2) - 5(p + 2) = 0 \] Now we can factor by grouping: \[ (3p - 5)(p + 2) = 0 \] Setting each factor equal to zero gives us: 1. \( 3p - 5 = 0 \) → \( p = \frac{5}{3} \) 2. \( p + 2 = 0 \) → \( p = -2 \) Thus, the solutions to the equation are \( p = \frac{5}{3} \) and \( p = -2 \).