A rectangular garden has an area expressed as a quadratic expression:

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Here are the solutions to the problems:

1. A rectangular garden has an area expressed as a quadratic expression:
$A = x^2 + 7x + 6$ .
Factorize the expression.

Find two numbers that multiply to 6 and add up to 7.
These numbers are 1 and 6.
Rewrite the middle term using these two numbers.
$x^2 + x + 6x + 6$
Factor by grouping.
$x(x + 1) + 6(x + 1)$
Factor out the common binomial factor.
$\boxed{(x + 1)(x + 6)}$
2. A carpenter calculates the height of a storage shelf using the expression
$x^2 - 1$ .
Factorize completely.
Recognize the expression as a difference of squares, $a^2 - b^2 = (a - b)(a + b)$ , where $a = x$ and $b = 1$ .
Apply the difference of squares formula.
$x^2 - 1^2 = \boxed{(x - 1)(x + 1)}$
3. A student multiplies two numbers and expresses the result in index notation:
$5 \times 5 \times 5 \times 5 \times 5$ using indices.
Count how many times the base number 5 is multiplied by itself.
It is multiplied 5 times.
Write the expression in index notation.
$\boxed{5^5}$
4. A chemical storage label shows an expression
$\frac{a^5}{a^7}$
using laws of indices.
Apply the law of indices for division, which states that $\frac{a^m}{a^n} = a^{m-n}$ .
$\frac{a^5}{a^7} = a^{5-7}$
Subtract the exponents.
$= a^{-2}$
Express with a positive exponent, using the rule $a^{-n} = \frac{1}{a^n}$ .
$= \boxed{\frac{1}{a^2}}$

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