Convolve |n| and 2n sequences

Write both sequences with their sample positions. (The second signal is called $y[n]$ in the question; to avoid confusion it is renamed $h[n]$ and the result is $c[n]=x[n]*h[n]$.)

$$x[n]=|n|:\quad x[-2]=2,\ x[-1]=1,\ x[0]=0,\ x[1]=1,\ x[2]=2$$ $$h[n]=2n:\quad h[2]=4,\ h[3]=6,\ h[4]=8,\ h[5]=10$$

The result starts at $n=-2+2=0$, ends at $n=2+5=7$, and has $5+4-1=8$ samples.

Tabular method

Rows are $x[k]\,h[n-k]$ for each $k$:

Example check: $c[3]=x[-2]h[5]+x[-1]h[4]+x[0]h[3]+x[1]h[2]=20+8+0+4=32$. Sum check: $\sum c = 168 = (\sum x)(\sum h)=6\times 28$.

Answer: $c[n]=\{8,16,22,32,24,20,26,20\}$ for $n=0,1,\dots,7$, zero elsewhere.

Plot

    c[n]
     32             o
     26             |           o
     24             |   o       |
     22         o   |   |       |
     20         |   |   |   o   |   o
     16     o   |   |   |   |   |   |
      8 o   |   |   |   |   |   |   |
      0 +---+---+---+---+---+---+---+--> n
        0   1   2   3   4   5   6   7