Question

factor ( x^4 + 10x^2 + 25 ) completely. all factors in your answer should have integer coefficients.

Explanation:

Step1: Recognize the quadratic form

Let \( y = x^2 \), then the expression becomes \( y^2 + 10y + 25 \).

Step2: Apply perfect square formula

The perfect square trinomial formula is \( a^2 + 2ab + b^2=(a + b)^2 \). Here, \( a = y \), \( 2ab = 10y \) (so \( b = 5 \)) and \( b^2 = 25 \). So \( y^2 + 10y + 25=(y + 5)^2 \).

Step3: Substitute back \( y = x^2 \)

Substitute \( y = x^2 \) into \( (y + 5)^2 \), we get \( (x^2 + 5)^2 \).

Answer:

\((x^2 + 5)^2\)