Question

what is the factored form of this expression?
$x^{2}+9x + 16$
a. $(x + 3)^{2}$
b. $(x + 4)^{2}$
c. the expression cannot be factored.
d. $(x - 4)(x + 4)$

Explanation:

Step1: Recall factoring quadratic form

For a quadratic expression \(ax^{2}+bx + c\) (\(a = 1\), \(b=9\), \(c = 16\) here), we need to find two numbers \(m\) and \(n\) such that \(m + n=b\) and \(mn=c\).

Step2: Check factor - pair possibilities

We look for two numbers that add up to 9 and multiply to 16. The factor - pairs of 16 are \((1,16)\), \((2,8)\), \((4,4)\). None of these pairs add up to 9.

Step3: Check special - form factoring

For \((x + a)^{2}=x^{2}+2ax + a^{2}\).

• For option A, \((x + 3)^{2}=x^{2}+6x + 9

eq x^{2}+9x + 16\).

• For option B, \((x + 4)^{2}=x^{2}+8x + 16

eq x^{2}+9x + 16\).

• For option D, \((x - 4)(x + 4)=x^{2}-16

eq x^{2}+9x + 16\).

Answer:

C. The expression cannot be factored.