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$\frac{9(y^4+6y^3+9y^2)}{y^2+6y+9} = \frac{9y^{2}( y^{2} + 6y + 9)}{ y^{2} + 6y + 9} = 9y^{2} = \frac{9}{6}(2014) = \boxed{3021}$

Simplify:

$\frac{9(y^{4}+6y^{3}+9y^{2})} {y^{2}+6y+9}$

$= \frac{9y^{2}(y^{2}+6y+9)}{y^{2}+6y+9}$

$=9y^2$

Make $6y^{2}$ into $9y^{2}$ to find the value of $9y^{2}$ .

$6y^{2}=2014\\2\cdot3y^{2}=2014\\3y^{2}=1007$

$3\cdot3y^{2}=3\cdot1007\\9y^{2}=3021$

9(y^4+6y^3+9y^2)=9(y^2+3y)^2 Factoring out the y^2, we obtain 9y^2(y+3)^2.

Factorising y^2+6y+9, we obtain (y+3)^2.

9(y^4+6y^3+9y^2)/y^2+6y+9 = 9y^2(y+3)^2/(y+3)^2 =9y^2

Since 6y^2=2014, 9y^2=2014÷2×3

And we get 3021 as the final answer.

9y^2(y^2+6y+9)*(9y^2+6y+9) 9y^2=6y^2+3y^2=3021

$\frac{9(y^4+6y^3+9y^2)}{y^2+6y+9}=9y^2$

$y^2=\frac{6y^2}{6}=\frac{2014}{6} \implies 9y^2=9\frac{2014}{6}=\frac{3 \times 2014}{6}=\boxed{\large{3021}}$

Factor the numerator:

$\dfrac{9(y^4+6y^3+9y^2)}{y^2+6y+9} = \dfrac{9y^2(y^2+6y+9)}{y^2+6y+9}=9y^2$

From $6y^2=2014$ , multiply both sides by $\dfrac{9}{6}$ to make the coefficient of $y^2$ equal to $9$ .

$\left(\dfrac{9}{6}\right)(6y^2)=2014\left(\dfrac{9}{6}\right)$

$\color{#D61F06}\large \boxed{9y^2=3021}$

9y^2 ( 9 + 6y + y^2 ) / ( 9 + 6y + y^2 )

=

9y^2 = 3 * 3y^2 & 3y^2 = 2014/2 = 1007

3 * 3y^2 = 3 * 1007

= 3021

9y^2(y^2+6y+9) / (y^2+6y+9) = 9*2014/6 = 3021

y^2=2014/6 from the equation taking y^2 common the equation reduce to 9y^2 =9*2014/6=3021.