To determine the amount of border material Jessica will need to protect the solar panels, we should calculate the perimeter of the panels. The given dimensions are:
- Width ([tex]\(W\)[/tex]): [tex]\(4c^2 + 5c\)[/tex]
- Length ([tex]\(L\)[/tex]): [tex]\(7c - 3\)[/tex]
The perimeter ([tex]\(P\)[/tex]) of a rectangle is given by the formula:
[tex]\[ P = 2 \times (W + L) \][/tex]
Now, let's substitute the expressions for width and length into this formula:
[tex]\[ P = 2 \times \left( (4c^2 + 5c) + (7c - 3) \right) \][/tex]
Combine like terms inside the parentheses:
[tex]\[ P = 2 \times \left( 4c^2 + 5c + 7c - 3 \right) \][/tex]
[tex]\[ P = 2 \times \left( 4c^2 + 12c - 3 \right) \][/tex]
Next, distribute the 2 to each term inside the parentheses:
[tex]\[ P = 2 \times 4c^2 + 2 \times 12c + 2 \times (-3) \][/tex]
[tex]\[ P = 8c^2 + 24c - 6 \][/tex]
Therefore, the standard form expression that represents the amount of border material Jessica will need is:
[tex]\[ \boxed{8c^2 + 24c - 6} \][/tex]
