On a screen giving you a few seconds per question, long multiplication is too slow. These techniques are.
Round and correct
The workhorse. Round one factor to something easy, then adjust.
47 times 8: 50 times 8 = 400, minus 3 times 8 = 24, gives 376.
98 times 34: 100 times 34 = 3400, minus 2 times 34 = 68, gives 3332.
Works whenever a factor is near a round number, which is most of the time.
Difference of squares
If two numbers are equidistant from an easy midpoint, use (a-b)(a+b) = a^2 - b^2.
47 times 53: midpoint 50, offset 3. So 2500 - 9 = 2491.
38 times 42: 1600 - 4 = 1596.
Instant once you spot it, and interviewers use these numbers deliberately.
Squares ending in 5
Take the tens digit n, compute n(n+1), append 25.
65^2: 6 times 7 = 42, so 4225. 85^2: 8 times 9 = 72, so 7225.
No exceptions, works for any size.
Multiply by 11
Add adjacent digits inward.
34 times 11: 3, then 3+4=7, then 4, giving 374. 87 times 11: 8, then 8+7=15 (carry), then 7, giving 957.
Halving and doubling
Halve one factor and double the other to reach an easier pair.
35 times 16 becomes 70 times 8 = 560, or 140 times 4, or 280 times 2.
Especially good when one factor is even and the other ends in 5.
Splitting
Break one factor into parts you know.
23 times 47 = 20 times 47 + 3 times 47 = 940 + 141 = 1081.
Slower than the others but universal - the fallback when no pattern fits.
The real skill
Not knowing the tricks - recognising which applies within a second. That only comes from volume. Work the mental math drill daily rather than in long occasional sessions; this is a motor skill and it consolidates overnight.
See also cracking speed mental math tests.