To find the factored form of the expression [tex]\( 8x^2 + 12x \)[/tex], let's follow these steps:
1. Identify the Greatest Common Factor (GCF):
We start by identifying the GCF for the terms in the given expression. The expression is [tex]\( 8x^2 + 12x \)[/tex].
- The coefficients are 8 and 12. The GCF of 8 and 12 is 4.
- The variable part is [tex]\( x^2 \)[/tex] and [tex]\( x \)[/tex]. The GCF here is [tex]\( x \)[/tex].
Thus, the GCF of the entire expression is [tex]\( 4x \)[/tex].
2. Factor Out the GCF:
Now we factor out [tex]\( 4x \)[/tex] from each term in the expression:
[tex]\[
8x^2 + 12x = 4x( \frac{8x^2}{4x} + \frac{12x}{4x} )
\][/tex]
Simplifying each term inside the parentheses:
- [tex]\(\frac{8x^2}{4x} = 2x \)[/tex]
- [tex]\(\frac{12x}{4x} = 3 \)[/tex]
So, we have:
[tex]\[
8x^2 + 12x = 4x(2x + 3)
\][/tex]
3. Verify with Given Options:
We should now verify the correct matching option:
Option 1: [tex]\(4(4x^2 + 8x)\)[/tex] \\
Option 2: [tex]\(4x(2x + 3)\)[/tex] \\
Option 3: [tex]\(8x(x + 4)\)[/tex] \\
Option 4: [tex]\(8x(x^2 + 4)\)[/tex]
From our factorization, [tex]\( 4x(2x + 3) \)[/tex] exactly matches Option 2.
Hence, the factored form of [tex]\( 8x^2 + 12x \)[/tex] is [tex]\( \boxed{4x(2x + 3)} \)[/tex] which corresponds to option 2.
