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. Step 1: Find two numbers that multiply to 6 and add up to 7. These numbers are 1 and 6. Step 2: Rewrite the middle term using these two numbers. $$x^2 + x + 6x + 6$$ Step 3: Factor by grouping. $$x(x + 1) + 6(x + 1)$$ Step 4: 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. Step 1: Recognize the expression as a difference of squares, $a^2 - b^2 = (a - b)(a + b)$, where $a = x$ and $b = 1$. Step 2: 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. Step 1: Count how many times the base number 5 is multiplied by itself. It is multiplied 5 times. Step 2: 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. Step 1: Apply the law of indices for division, which states that $\frac{a^m}{a^n} = a^{m-n}$. $$\frac{a^5}{a^7} = a^{5-7}$$ Step 2: Subtract the exponents. $$ = a^{-2}$$ Step 3: Express with a positive exponent, using the rule $a^{-n} = \frac{1}{a^n}$. $$ = \boxed{\frac{1}{a^2}} $$ That's 2 down. 3 left today — send the next one.