(b) Write your answer in the following form: Quotient + Remainder \[ \frac{6 x^{4}+19 x^{3}-4 x+15}{x+3}=\square+\frac{\square}{x+3} \]

Let's perform polynomial long division with \( 6x^4 + 19x^3 - 4x + 15 \) divided by \( x + 3 \). 1. Divide the leading term: \( \frac{6x^4}{x} = 6x^3 \). 2. Multiply \( 6x^3 \) by \( x + 3 \): \( 6x^4 + 18x^3 \). 3. Subtract: \[ (19x^3 - 18x^3) - 4x + 15 = x^3 - 4x + 15. \] 4. Repeat the process with \( x^3 \): \( \frac{x^3}{x} = x^2 \). 5. Multiply \( x^2 \) by \( x + 3 \): \( x^3 + 3x^2 \). 6. Subtract: \[ (x^3 - x^3) + (19x^3 - 3x^2) - 4x + 15 = -3x^2 - 4x + 15. \] 7. Now repeat with \( -3x^2 \): \( \frac{-3x^2}{x} = -3x \). 8. Multiply \( -3x \) by \( x + 3 \): \( -3x^2 - 9x \). 9. Subtract: \[ (-3x^2 + 3x^2) + (19x - (-9x)) + 15 = 5x + 15. \] 10. Next, \( \frac{5x}{x} = 5 \). 11. Multiply \( 5 \) by \( x + 3 \): \( 5x + 15 \). 12. Subtract: \[ (5x - 5x) + (15 - 15) = 0. \] So, the polynomial division gives us a quotient of \( 6x^3 + x^2 - 3x + 5 \) and a remainder of \( 0 \). Finally, we can express the original division in the required form: \[ \frac{6x^{4}+19x^{3}-4x+15}{x+3}= 6x^3 + x^2 - 3x + 5 + \frac{0}{x+3}. \] Thus, the answer is: \[ 6x^3 + x^2 - 3x + 5 + \frac{0}{x+3}. \]