Use the long division method to find the result when \( 3 x^{3}+16 x^{2}+10 x+25 \) is divided by \( x+5 \).

To use long division to divide \( 3x^{3} + 16x^{2} + 10x + 25 \) by \( x + 5 \), we start by setting it up as follows: 1. **Divide the leading terms**: Divide \( 3x^3 \) by \( x \), which gives \( 3x^2 \). 2. **Multiply and subtract**: Multiply \( 3x^2 \) by \( x + 5 \) to get \( 3x^3 + 15x^2 \). Subtract this from the original polynomial: \[ (3x^3 + 16x^2 + 10x + 25) - (3x^3 + 15x^2) = 1x^2 + 10x + 25 \] 3. **Repeat the process**: Next, divide \( 1x^2 \) by \( x \) to get \( 1x \). Multiply \( 1x \) by \( x + 5 \): \[ 1x(x + 5) = x^2 + 5x \] Subtract: \[ (1x^2 + 10x + 25) - (1x^2 + 5x) = 5x + 25 \] 4. **Continue dividing**: Divide \( 5x \) by \( x \) to get \( 5 \). Multiply \( 5 \) by \( x + 5 \): \[ 5(x + 5) = 5x + 25 \] Subtract: \[ (5x + 25) - (5x + 25) = 0 \] Since the remainder is \( 0 \), we conclude that the result of the division is: \[ 3x^2 + x + 5 \] Thus, when \( 3x^3 + 16x^2 + 10x + 25 \) is divided by \( x + 5 \), the quotient is \( 3x^2 + x + 5 \).