Factorization - Common Factor: Extracting a Common Factor from a group of numbers following decomposition

To factor the expression $4x^2 + 16x$, we start by identifying the greatest common factor (GCF) of $4x^2$ and $16x$. The GCF is $4x$. We factor out $4x$ from each term:

$4x^2+16x=\blue4\cdot \orange x\cdot x+\blue4\cdot4\cdot \orange x$.

This simplifies the expression to $4x(x + 4)$.

To factor the expression $4x^2 + 2x$, first notice that each term shares a common factor of $2x$: $\blue 2\cdot2\cdot \orange x\cdot x+\blue 2\cdot \orange x$

Start by factoring out $2x$:

$4x^2 + 2x = 2x(2x + 1)$

Thus, the factored expression is $2x(2x + 1)$, as $2 \cdot x$ is the common factor.

To factor the expression $5x^2 + 10x$, first identify the common factor in each term. Here, both terms share a $5x$ factor.

$\blue 5\cdot \orange x\cdot x+\blue5\cdot2\cdot\orange x$

Next, factor out $5x$ from each term:

$5x^2 + 10x = 5x(x + 2)$

The common factor extracted is $5x(x + 2)$, making it the simplified expression.

For the expression $6x^2 + 3x$, the greatest common factor is $3x$: $\blue 3\cdot 2\cdot \orange x\cdot x+\blue3\cdot \orange x$

When you factor out $3x$, the expression becomes:

$6x^2 + 3x = 3x(2x + 1)$

This expresses the original expression in its factored form, with $3x(2x + 1)$ being the simplest form.

To factor the expression $3x^2 + 12x$, we start by looking for the greatest common factor (GCF) of the terms $3x^2$ and $12x$. The GCF is $3x$. We can factor out $3x$ from each term:

$3x^2+12x=\blue3\cdot \orange x\cdot x+\blue 3\cdot4\cdot \orange x$.

This allows us to write the expression as $3x(x + 4)$.

We start with the expression $3y + 6y$.

First, we notice that both terms share a common factor of $y$.

So, we factor out $y$ from each term:

$3y = y\cdot 3$ and $6y = y\cdot 6$.

We can also see that $6$ can be factored to $2\cdot3$.
Now when we look at the expression we can see that both $y$ and $3$ are the common factor: $\blue3\cdot \orange y+2\cdot\blue 3\cdot \orange y$

This allows us to rewrite the expression as $3y\left(1+2\right)$, as nothing is left from the first term, and so we keep there a $1$, and $2$ is left from the second term.

Thus, the factored form is $3y\left(1+2\right)$

To factor the expression $5x^2 + 25x$, we look for the greatest common factor (GCF) of the terms $5x^2$ and $25x$. The GCF is $5x$. We factor out $5x$ from each term:

$5x^2+25x=\blue5\cdot \orange x\cdot x+\blue 5\cdot5\cdot \orange x$.

This results in the expression $5x(x + 5)$.

We start with the expression $9a+8a$.

First, we notice that both terms share a common factor of $a$.

So, we factor out$a$ from each term:

$9a=a\cdot9$ and $8a = a\cdot 8$.

This allows us to rewrite the expression as $a(9+8)$.

Thus, the factored form is $a\left(9+8\right)$.