An expression for the area of a shape is \( 6(a+5) \mathrm{cm}^{2} \). (a) the area when \( a=-2 \). (b) the value of \( a \) when the area is \( 54 \mathrm{~cm}^{2} \).

To solve the problems, we will use the expression for the area given as \( 6(a+5) \mathrm{cm}^{2} \). ### (a) The area when \( a = -2 \) Substituting \( a = -2 \) into the area expression: \[ \text{Area} = 6(-2 + 5) \] Calculating inside the parentheses: \[ -2 + 5 = 3 \] Now substituting back into the area expression: \[ \text{Area} = 6 \times 3 = 18 \mathrm{cm}^{2} \] So, the area when \( a = -2 \) is \( \boxed{18 \mathrm{cm}^{2}} \). ### (b) The value of \( a \) when the area is \( 54 \mathrm{cm}^{2} \) We set the area expression equal to \( 54 \): \[ 6(a + 5) = 54 \] To solve for \( a \), first divide both sides by 6: \[ a + 5 = \frac{54}{6} \] Calculating the right side: \[ \frac{54}{6} = 9 \] Now, we have: \[ a + 5 = 9 \] Subtracting 5 from both sides gives: \[ a = 9 - 5 = 4 \] Thus, the value of \( a \) when the area is \( 54 \mathrm{cm}^{2} \) is \( \boxed{4} \).