Joint probability density function with complicated exponential function

Question

There is given the following function:

$f(x,y) = ce^{-4(x+1)^2-6(x+1)y-9y^2}$

In the task I am asked to do the following things

1. Find $c$

2. Marginal distribution of $X$ and $Y$

3. Probability density function of $T = x^2 + 3x +1$

I am not really able or have any idea how to solve this task. The problem is the function that does not look nice. I have already done other similar tasks, but functions in those ones were not so complicated as here.

I have heard that there is a need to use some matrix and then the determinant of it, but I can't see what values should be take into account when creating this matrix.

Answer 1

1. To find c, calculate the integral over all x and y of f(x,y). Adjust c so the integral = 1.

2. Calculate the integral over x to get the density function for Y and similarly integral over y to get the density for X.

3. Statement looks funny. Why T=x+3x+1 (=4x+1)?