Certainly! Let's expand the given expression and identify the corresponding coefficients step-by-step.
We start with the given binomials:
[tex]\[
(x+2)(x+3)
\][/tex]
To expand this, we apply the distributive property (also known as the FOIL method when dealing with binomials):
1. First: Multiply the first terms of each binomial.
[tex]\[
x \cdot x = x^2
\][/tex]
2. Outside: Multiply the outer terms.
[tex]\[
x \cdot 3 = 3x
\][/tex]
3. Inside: Multiply the inner terms.
[tex]\[
2 \cdot x = 2x
\][/tex]
4. Last: Multiply the last terms.
[tex]\[
2 \cdot 3 = 6
\][/tex]
Now, we combine all these results:
[tex]\[
x^2 + 3x + 2x + 6
\][/tex]
Next, combine like terms (the [tex]\(x\)[/tex] terms):
[tex]\[
x^2 + (3x + 2x) + 6 = x^2 + 5x + 6
\][/tex]
So, the expanded form of the expression [tex]\((x+2)(x+3)\)[/tex] is:
[tex]\[
x^2 + 5x + 6
\][/tex]
From the expanded expression, we can identify the coefficients corresponding to the powers of [tex]\(x\)[/tex]:
- The coefficient of [tex]\(x^2\)[/tex] is [tex]\(1\)[/tex],
- The coefficient of [tex]\(x\)[/tex] is [tex]\(5\)[/tex],
- The constant term (coefficient of [tex]\(x^0\)[/tex]) is [tex]\(6\)[/tex].
Therefore, the expression [tex]\((x+2)(x+3)\)[/tex] expands to:
[tex]\[
x^2 + 5x + 6
\][/tex]
This means the answer to fill in the blanks is:
[tex]\[
\begin{array}{l}(x+2)(x+3) \\ x^2+[5]x+6\end{array}
\][/tex]
So, the missing coefficients are [tex]\(5\)[/tex] and [tex]\(6\)[/tex].
