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Problem 358

# Find A) $$4 a^{2} b+2 a b$$ B) $$9 a b^{2} c-6 a^{2} c+12 a c$$ C) $$\mathrm{ac}+\mathrm{bc}+\mathrm{ad}+\mathrm{bd}$$

Expert verified
The simplified expressions are: A) $$2ab(2a+1)$$ B) $$3ac(3b^2-2a+4)$$ C) $$(a+b)(c+d)$$
See the step by step solution

## Step 1: Identify common factors

In this expression, we can see that both terms have $$a$$ and $$b$$ as their common factors.

## Step 2: Factor out the common factors

We can factor out $$ab$$ from both terms: $ab(4a+2)$

## Step 3: Simplify the expression

Now, we can simplify the expression further by factoring out the common factor $$2$$ from the parentheses: $2ab(2a+1)$ So, the simplified expression for A is: $2ab(2a+1)$ B) $$9ab^2c-6a^2c+12ac$$

## Step 1: Identify common factors

In this expression, we can see that all three terms have $$a$$ and $$c$$ as their common factors.

## Step 2: Factor out the common factors

We can factor out $$ac$$ from all three terms: $ac(9b^2-6a+12)$

## Step 3: Simplify the expression

Now, we can simplify the expression further by factoring out the common factor $$3$$ from the parentheses: $3ac(3b^2-2a+4)$ So, the simplified expression for B is: $3ac(3b^2-2a+4)$ C) $$ac+bc+ad+bd$$

## Step 1: Identify terms with common factors

In this expression, we can see that the first and third terms have $$a$$ as a common factor, and the second and fourth terms have $$b$$ as a common factor.

## Step 2: Factor out the common factors

We can factor out $$a$$ from the first and third terms and $$b$$ from the second and fourth terms: $a(c+d)+b(c+d)$

## Step 3: Simplify the expression

Now, we can simplify the expression further by factoring out the common factor $$(c+d)$$: $(a+b)(c+d)$ So, the simplified expression for C is: $(a+b)(c+d)$

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