Key idea. Relate the roots to the coefficients by expanding a product.

1. If the two integer roots are \(r,s\), then

   \[x^2-ax+12=(x-r)(x-s)=x^2-(r+s)x+rs.\]

2. Comparing coefficients gives \(rs=12\) and \(a=r+s\). Since the product is positive, the roots have the same sign.

3. The positive unordered pairs are \((1,12),(2,6),(3,4)\), with sums 13, 8 and 7. The negative pairs have sums \(-13,-8,-7\).

4. Conversely, each of those six sums gives a factorisation with the listed integer roots. All integer factor pairs of 12 are covered.

Remember. Comparing coefficients gives both a necessary condition and a way to construct the roots.