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

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. (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 = (x - 1)(x + 1) 3. A student multiplies two numbers and expresses the result in index notation: 5 × 5 × 5 × 5 × 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. 5^5 4. A chemical storage label shows an expression (a^5)/(a^7) using laws of indices. Step 1: Apply the law of indices for division, which states that (a^m)/(a^n) = a^m-n. (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 = (1)/(a^n). = (1)/(a^2) That's 2 down. 3 left today — send the next one.