To subtract $(a^2 + 5ab - 6b^2)$ from $(a^2 + b^2)$, write the expression as $(a^2 + b^2) - (a^2 + 5ab - 6b^2)$. Distribute the subtraction by changing the sign of each term in the second polynomial to get $a^2 + b^2 - a^2 - 5ab + 6b^2$. Next, rearrange the terms to group the like terms together: $a^2 - a^2 - 5ab + b^2 + 6b^2$. Combine the like terms to find the difference: $-5ab + 7b^2$.