Ask two questions before declaring bias
In the relevant linear-model setting, ask whether omitted z affects y and whether z is correlated with included x. Saying that other factors exist is insufficient to identify omitted-variable bias in the coefficient of interest.
The open textbook Introduction to Econometrics with R explains these conditions. A larger sample does not automatically correct the underlying specification problem.
Derive an original population example
Suppose y=5+2x+3z+u with zero conditional mean of u given x and z. Let the population linear projection of z on x be z=1+0.5x+v, with v uncorrelated with x.
Substitution gives y=8+3.5x+(3v+u). The short regression has population slope 3.5 rather than the full-model coefficient 2. The difference is 3×0.5=1.5. This is an invented teaching mechanism, not an empirical finding.
Reason about signs separately
Keep the coefficient on z at 3 but change its projection slope on x to −0.5. The omitted contribution becomes −1.5 and the short-regression population slope becomes 0.5. Explain both signs rather than memorising a direction.
If either component is zero, this particular bias term is zero. More controls are not a universal cure: justify their theoretical role, measurement and timing instead of selecting variables to obtain a desired significance result.
State the limits of interpretation
Report the association first and list assumptions needed for a causal interpretation separately. A coefficient changing after controls are added does not prove that all confounding has been removed.
Self-check: a coefficient on z of −4 and projection slope of −0.25 give a bias term of +1. Explain each negative sign. Use permitted teaching material and independent reasoning rather than outsourced assessed analysis.
References
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