Sure, let's factorize the expression [tex]\(8 p x^3 + 12 p x^2\)[/tex] step by step.
1. Identify the common factors:
First, we observe that both terms in the expression [tex]\(8 p x^3\)[/tex] and [tex]\(12 p x^2\)[/tex] share common factors:
- Both terms have the variable [tex]\(p\)[/tex].
- Both terms have the variable [tex]\(x\)[/tex], specifically with powers [tex]\(x^3\)[/tex] and [tex]\(x^2\)[/tex].
- Both terms are divisible by 4.
2. Factor out the greatest common factor (GCF):
The GCF for the numerical coefficients 8 and 12 is 4. Both terms contain [tex]\(p\)[/tex] and [tex]\(x^2\)[/tex] as the highest common power of [tex]\(x\)[/tex]. Thus, the GCF is [tex]\(4 p x^2\)[/tex].
[tex]\[
4 p x^2 \times \left(\frac{8 p x^3}{4 p x^2} + \frac{12 p x^2}{4 p x^2}\right)
\][/tex]
3. Simplify inside the parentheses:
Simplify the terms inside the parentheses after factoring out [tex]\(4 p x^2\)[/tex]:
[tex]\[
4 p x^2 \left(2 x + 3\right)
\][/tex]
4. Combine the terms:
Write the final, factorized form:
[tex]\[
4 p x^2 (2 x + 3)
\][/tex]
So, the expression [tex]\(8 p x^3 + 12 p x^2\)[/tex] factorizes to:
[tex]\[
4 p x^2 (2 x + 3)
\][/tex]
This is our final, fully factorized expression.
