Answer
$${\text{ellipse}}$$
Work Step by Step
$$\eqalign{
& 4{x^2} - 8x + 9{y^2} + 36y = - 4 \cr
& {\text{Factor }} \cr
& \left( {4{x^2} - 8x} \right) + \left( {9{y^2} + 36y} \right) = 4 \cr
& 4\left( {{x^2} - 2x} \right) + 9\left( {{y^2} + 4y} \right) = 4 \cr
& {\text{Complete the square}} \cr
& 4\left( {{x^2} - 2x + 1} \right) + 9\left( {{y^2} + 4y + 4} \right) = 4 + 4\left( 1 \right) + 9\left( 4 \right) \cr
& 4{\left( {x - 1} \right)^2} + 9{\left( {y + 2} \right)^2} = 44 \cr
& {\text{Divide both sides by 44}} \cr
& \frac{{{{\left( {x - 1} \right)}^2}}}{{11}} + \frac{{{{\left( {y + 2} \right)}^2}}}{{44/9}} = 1 \cr
& {\text{The equation is written in the form }}\frac{{{{\left( {x - h} \right)}^2}}}{{{a^2}}} + \frac{{{{\left( {y - k} \right)}^2}}}{{{b^2}}} = 1\,\,\, \cr
& {\text{Therefore,}} \cr
& {\text{The graph of the equation is an ellipse}} \cr} $$
