To perform the division of the polynomial \( \frac{y^2 + 2y + 11}{y + 6} \), we can use polynomial long division.
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Divide the leading term of the numerator \( y^2 \) by the leading term of the denominator \( y \):
\[ y^2 \div y = y \]
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Multiply the entire divisor \( y + 6 \) by this result \( y \):
\[ y(y + 6) = y^2 + 6y \]
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Subtract this from the original numerator:
\[ (y^2 + 2y + 11) - (y^2 + 6y) = 2y + 11 - 6y = -4y + 11 \]
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Now, we take the new polynomial \(-4y + 11\) and repeat the process. Divide the leading term \(-4y\) by \(y\):
\[ -4y \div y = -4 \]
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Multiply the entire divisor \(y + 6\) by \(-4\):
\[ -4(y + 6) = -4y - 24 \]
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Subtract this from \(-4y + 11\):
\[ (-4y + 11) - (-4y - 24) = 11 + 24 = 35 \]
Now, we are left with a remainder of \(35\).
Combining all parts, the result of the division is:
\[ y - 4 + \frac{35}{y + 6} \]
So, the final answer is:
\[ \frac{y^2 + 2y + 11}{y + 6} = y - 4 + \frac{35}{y + 6} \]