Use factoring to solve the quadratic equation. Check by substitution or by using a graphing utility and \( 5 x^{2}=38 x+16 \) The solution set is (Use commas to separate answers as needed. Type each solution only once.)

To solve the equation \( 5 x^{2} = 38 x + 16 \), first rearrange it into standard form: \[ 5x^2 - 38x - 16 = 0. \] Next, factor the quadratic. To do this effectively, we need two numbers that multiply to \( 5 \times -16 = -80 \) and add up to \(-38\). Those numbers are \(-40\) and \(2\). Rewrite the quadratic: \[ 5x^2 - 40x + 2x - 16 = 0. \] Now, factor by grouping: \[ 5x(x - 8) + 2(x - 8) = 0 \] Factor out the common term \((x - 8)\): \[ (5x + 2)(x - 8) = 0. \] Set each factor to zero: 1. \( 5x + 2 = 0 \) ⟹ \( 5x = -2 \) ⟹ \( x = -\frac{2}{5} \) 2. \( x - 8 = 0 \) ⟹ \( x = 8 \) Thus, the solution set is: \(-\frac{2}{5}, 8\) Now, let's check by substitution: 1. For \( x = -\frac{2}{5} \): \[ 5(-\frac{2}{5})^2 = 38(-\frac{2}{5}) + 16, \] \[ 5 \times \frac{4}{25} = -\frac{76}{5} + 16, \] \[ \frac{4}{5} = -\frac{76}{5} + \frac{80}{5} = \frac{4}{5} \] (True) 2. For \( x = 8 \): \[ 5(8^2) = 38(8) + 16, \] \[ 5 \times 64 = 304 + 16, \] \[ 320 = 320 \] (True) Both solutions are accurate!