Convolve {3, 1, 5} and {1, 4, −2, 3}

The convolution sum is $y[n]=\sum_k x[k]h[n-k]$. Since the origin is not marked, take the first sample of each sequence at $n=0$: $x[n]=\{3,1,5\}$ for $n=0,1,2$ and $h[n]=\{1,4,-2,3\}$ for $n=0,\dots,3$.

Length of $y$ = $3+4-1=6$, running from $n=0+0=0$ to $n=2+3=5$.

Tabular (multiplication) method

Each row is $x[k]$ times the whole of $h$, shifted by $k$:

Check of each value

$$\begin{aligned} y[0]&=3(1)=3\\ y[1]&=3(4)+1(1)=13\\ y[2]&=3(-2)+1(4)+5(1)=3\\ y[3]&=3(3)+1(-2)+5(4)=27\\ y[4]&=1(3)+5(-2)=-7\\ y[5]&=5(3)=15 \end{aligned}$$

Check: $\sum y = 54 = (\sum x)(\sum h) = 9\times 6$. Correct.

Answer: $y[n]=\{3,\ 13,\ 3,\ 27,\ -7,\ 15\}$ for $n=0$ to $5$ (↑ at 3).