To determine the possible dimensions of the rectangular painting given the trinomial expression [tex]\( x^2 + 4x - 60 \)[/tex], we need to factor the trinomial.
1. Identify the trinomial: The given polynomial is [tex]\( x^2 + 4x - 60 \)[/tex].
2. Form a factor pair: We need to write the trinomial in the form [tex]\((x + a)(x + b)\)[/tex], where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are constants. To do so, we look for two numbers whose product is equal to the constant term [tex]\(-60\)[/tex] and whose sum is equal to the coefficient of [tex]\(x\)[/tex] (which is 4).
3. Find the appropriate numbers:
- We know that the two numbers [tex]\(a\)[/tex] and [tex]\(b\)[/tex] need to multiply to [tex]\(-60\)[/tex] and add up to 4.
- By trial or inspection, we find that [tex]\( -6 \)[/tex] and [tex]\( 10 \)[/tex] satisfy these conditions:
- [tex]\( -6 \times 10 = -60 \)[/tex]
- [tex]\( -6 + 10 = 4 \)[/tex]
4. Write out the factors:
- Given the numbers [tex]\(-6\)[/tex] and [tex]\(10\)[/tex], the trinomial can be factored as:
[tex]\[
(x - 6)(x + 10)
\][/tex]
5. Possible dimensions: The possible dimensions of the rectangular painting based on the factors we determined are:
- [tex]\( x - 6 \)[/tex]
- [tex]\( x + 10 \)[/tex]
Thus, the correct answer is:
a. [tex]\( x-6 \)[/tex] and [tex]\( x+10 \)[/tex].
