To determine what needs to be added to the expression [tex]\(a + 3b + 5ab\)[/tex] to obtain [tex]\(-2a + 6b - 2ab\)[/tex], we need to examine the differences in the coefficients of each term in the given expressions.
Let's denote the first expression as:
[tex]\[ E_1 = a + 3b + 5ab \][/tex]
And the second expression as:
[tex]\[ E_2 = -2a + 6b - 2ab \][/tex]
We want to find what must be added to [tex]\( E_1 \)[/tex] to result in [tex]\( E_2 \)[/tex].
Consider the general form of the left-hand side after adding something to [tex]\( E_1 \)[/tex]:
[tex]\[ (a + 3b + 5ab) + (ka + lb + mab) = -2a + 6b - 2ab \][/tex]
We now need to find constants [tex]\( k \)[/tex], [tex]\( l \)[/tex], and [tex]\( m \)[/tex] such that:
[tex]\[ (1 + k)a + (3 + l)b + (5 + m)ab = -2a + 6b - 2ab \][/tex]
Matching coefficients of the same terms on both sides we get:
1. For the coefficient of [tex]\(a\)[/tex]:
[tex]\[ 1 + k = -2 \][/tex]
Thus,
[tex]\[ k = -3 \][/tex]
2. For the coefficient of [tex]\(b\)[/tex]:
[tex]\[ 3 + l = 6 \][/tex]
Thus,
[tex]\[ l = 3 \][/tex]
3. For the coefficient of [tex]\(ab\)[/tex]:
[tex]\[ 5 + m = -2 \][/tex]
Thus,
[tex]\[ m = -7 \][/tex]
So, the expression that needs to be added to [tex]\( a + 3b + 5ab \)[/tex] is:
[tex]\[ -3a + 3b - 7ab \][/tex]
Therefore, the terms that should be added to [tex]\(a + 3b + 5ab\)[/tex] to obtain [tex]\(-2a + 6b - 2ab\)[/tex] are:
[tex]\[
\boxed{-3a + 3b - 7ab}
\][/tex]
