Let's solve the problem step by step.

Step 1: Identify the expressions for length and width.

Let the width of the rectangular garden be $w$. Since the length is 3 meters more than the width, the length can be expressed as $l = w + 3$.

Step 2: Write down the formula for the area of the rectangle.

The area of a rectangle is given by the formula: $\text{Area} = \text{Length} \times \text{Width}$ Substituting the expressions for length and width, we have: $\text{Area} = (w + 3) \times w$

Step 3: Set up the equation using the given area.

We are given that the area of the garden is $x^2 + 4x + 3$. So we equate the expression for the area to this polynomial: $w(w + 3) = x^2 + 4x + 3$

Step 4: Expand and simplify the equation.

Expanding the left side: $w^2 + 3w = x^2 + 4x + 3$

Step 5: Compare coefficients to solve for $w$.

For the two sides of the equation to be equal, the coefficients of corresponding terms must be equal.

• The coefficient of $w^2$ is 1 on both sides, so $w = x$.
• Substituting $w = x$ into the original expression, we have: $w^2 + 3w = x^2 + 3x$

Since the equation simplifies naturally, the width $w$ is equal to $x$.

Thus, the width of the garden is $x$ meters.

Would you like more details or have any questions?

Relative Questions

1. How do you determine the length if you know the width?
2. What would the dimensions be if the polynomial representing the area changed?
3. How do the coefficients in the polynomial relate to the dimensions of the rectangle?
4. Can this method be used for other shapes, and how would it differ?
5. How would you find the dimensions if the polynomial was a cubic equation?

Tip: Understanding how to set up equations based on geometric formulas is crucial for solving problems involving dimensions.