Factor the algebraic expressions: 3a2b + 4ab2, 7x + 35, and xy + 5y + 2x + 10.

The H.C.F. of the numerical coefficients (3 and 4) is 1.
The H.C.F. of the variable parts ($a^2b$ and $ab^2$) is $ab$.
So, the H.C.F. of the entire expression is $1 \times ab = ab$.

from each term.
Divide $3a^2b$ by $ab$: $\frac{3a^2b}{ab} = 3a$.
Divide $4ab^2$ by $ab$: $\frac{4ab^2}{ab} = 4b$.

The equivalent algebraic expression is $\boxed{ab(3a + 4b)}$.

To find an equivalent expression for $7x + 35$:

• First, identify the greatest common factor (GCF) of both terms, $7x$ and 35. The factors of $7x$ are $1, 7, x, 7x$. The factors of 35 are $1, 5, 7, 35$. The GCF is 7.
• Next, divide each term in the expression by the GCF. $7x \div 7 = x$ and $35 \div 7 = 5$.
• Finally, write the GCF outside a parenthesis and the results of the division inside the parenthesis. This gives the equivalent expression $7(x + 5)$.

For the expression $xy + 5y + 2x + 10$:
You can notice that there is no common factor (other than 1) for all four terms. However, the terms can be grouped into pairs that share common factors. The first two terms ($xy$ and $5y$) share a common factor of $y$, and the last two terms ($2x$ and 10) share a common factor of 2. This allows for factorization by grouping.