Question

which of the following is a factor of 2x^4 + 20x^3 + 50x^2?
2x^3
x^4
x + 4
x + 5

Explanation:

Step1: Factor out the GCF

First, find the greatest - common factor of the terms in the polynomial $2x^{4}+20x^{3}+50x^{2}$. The GCF of 2, 20, and 50 is 2, and the GCF of $x^{4},x^{3},x^{2}$ is $x^{2}$. So, $2x^{4}+20x^{3}+50x^{2}=2x^{2}(x^{2} + 10x+25)$.

Step2: Factor the quadratic

Factor the quadratic expression $x^{2}+10x + 25$. Using the formula $(a + b)^2=a^{2}+2ab + b^{2}$, where $a=x$ and $b = 5$ (since $2ab=2\times x\times5 = 10x$ and $a^{2}=x^{2}$, $b^{2}=25$), we get $x^{2}+10x + 25=(x + 5)^{2}$. So, $2x^{4}+20x^{3}+50x^{2}=2x^{2}(x + 5)^{2}$.

Answer:

$x + 5$