Enter the values for \( a \) and \( b \) that complete the sum: \( \frac{3}{x}+\frac{5}{x^{2}}=\frac{a x+b}{x^{2}} \)

We start with the equation: \[ \frac{3}{x}+\frac{5}{x^{2}}=\frac{a x+b}{x^{2}} \] **Step 1.** Express the left-hand side with a common denominator. The common denominator is \( x^2 \), so we write: \[ \frac{3}{x} = \frac{3x}{x^{2}} \] Thus: \[ \frac{3x}{x^{2}}+\frac{5}{x^{2}}=\frac{3x+5}{x^{2}} \] **Step 2.** Set the numerators equal since the denominators are the same: \[ 3x+5 = ax+b \] **Step 3.** Equating coefficients for like terms, we have: For the coefficient of \( x \): \[ 3 = a \] For the constant term: \[ 5 = b \] Therefore, the values are: \[ a=3, \quad b=5. \]