Output for h[n] = {1, 2, 2, 5}, x[n] = {1, 3}

Take $h[0]$ and $x[0]$ as the first samples (no origin is marked): $h[n]=\{1,2,2,5\}$, $n=0..3$ and $x[n]=\{1,3\}$, $n=0,1$.

The output $y[n]=x[n]*h[n]=\sum_k x[k]h[n-k]$ has $2+4-1=5$ samples, $n=0..4$.

Because $x[n]=\delta[n]+3\delta[n-1]$, the output is

$$y[n]=h[n]+3h[n-1]$$

Tabular method

Step values

$$\begin{aligned} y[0]&=1\\ y[1]&=2+3(1)=5\\ y[2]&=2+3(2)=8\\ y[3]&=5+3(2)=11\\ y[4]&=3(5)=15 \end{aligned}$$

Sum check: $\sum y=40=(\sum x)(\sum h)=4\times10$.

Answer: $y[n]=\{1,\ 5,\ 8,\ 11,\ 15\}$ for $n=0,\dots,4$ (↑ at 1).

    y[n]
     15                 o
     11             o   |
      8         o   |   |
      5     o   |   |   |
      1 o   |   |   |   |
      0 +---+---+---+---+--> n
        0   1   2   3   4