Convexity of $f(x) = x^TAx^Tx + C$

Under what conditions this function will be a convex function.$$f(x) = x^TAx^Tx + C$$

1. A is negative definite
2. A is positive semi-definite
3. A can be either positive or negative semi-definite
4. A is diagonal

My Try
Condition of convexity$$f(\lambda x + (1-\lambda)y) \le \lambda x + (1-\lambda)y$$Let $\lambda = \frac{1}{2}$

$$\left(\frac{x+y}{2}\right)^TA\left(\frac{x+y}{2}\right)^T\left(\frac{x+y}{2}\right) + C \le \left(\frac{x^TAx^Tx}{2}\right) +\frac{C}{2}+ \left(\frac{y^TAy^Ty}{2}\right) + \frac{C}{2}$$

$$\left(\frac{x}{2}\right)^TA\left(\frac{x+y}{2}\right)^T\left(\frac{x+y}{2}\right) + \left(\frac{y}{2}\right)^TA\left(\frac{x+y}{2}\right)^T\left(\frac{x+y}{2}\right) \le \left(\frac{x^TAx^Tx}{2}\right) + \left(\frac{y^TAy^Ty}{2}\right)$$

$$\left(\frac{x}{2}\right)^TA\left(\frac{x}{2}\right)^T\left(\frac{x+y}{2}\right) + \left(\frac{y}{2}\right)^TA\left(\frac{y}{2}\right)^T\left(\frac{x+y}{2}\right)+\left(\frac{x}{2}\right)^TA\left(\frac{y}{2}\right)^T\left(\frac{y}{2}\right) + \left(\frac{y}{2}\right)^TA\left(\frac{x}{2}\right)^T\left(\frac{x+y}{2}\right) \le \left(\frac{x^TAx^Tx}{2}\right) + \left(\frac{y^TAy^Ty}{2}\right)$$

$$\left(\frac{x}{2}\right)^TA\left(\frac{x}{2}\right)^T\left(\frac{x}{2}\right) + \left(\frac{y}{2}\right)^TA\left(\frac{y}{2}\right)^T\left(\frac{x}{2}\right)+\left(\frac{x}{2}\right)^TA\left(\frac{y}{2}\right)^T\left(\frac{x}{2}\right) + \left(\frac{y}{2}\right)^TA\left(\frac{x}{2}\right)^T\left(\frac{x}{2}\right) + \left(\frac{x}{2}\right)^TA\left(\frac{x}{2}\right)^T\left(\frac{y}{2}\right) + \left(\frac{y}{2}\right)^TA\left(\frac{y}{2}\right)^T\left(\frac{y}{2}\right)+\left(\frac{x}{2}\right)^TA\left(\frac{y}{2}\right)^T\left(\frac{y}{2}\right) + \left(\frac{y}{2}\right)^TA\left(\frac{x}{2}\right)^T\left(\frac{y}{2}\right) \le \left(\frac{x^TAx^Tx}{2}\right) + \left(\frac{y^TAy^Ty}{2}\right)$$Multiplying 8 both side$$x^TAx^Tx + y^TAy^Tx+x^TAy^Tx + y^TAx^Tx + x^TAx^Ty + y^TAy^Ty+x^TAy^Ty + y^TAx^Ty \le 4x^TAx^Tx + 4y^TAy^Ty$$$$y^TAy^Tx+x^TAy^Tx + y^TAx^Tx + x^TAx^Ty+x^TAy^Ty + y^TAx^Ty - 3x^TAx^Tx - 3y^TAy^Ty \le 0$$

Now how can I go further.